#33608

order	256
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

325 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,11,16)(7,17,22,42)(8,13,23,34)(9,26,28,54)(10,29,27,15)(12,30,33,20)(14,37,38,18)(19,39,45,61)(21,49,48,41)(24,52,50,67)(25,53,51,36)(31,55,57,46)(32,43,40,56)(35,60,59,44)(47,66,65,63)(58,64,62,68);(5,7,18,43,10,21,28,30,11,22,37,56,27,48,9,20)(6,12,26,41,15,32,38,42,16,33,54,49,29,40,14,17)(8,19,44,64,25,47,50,55,23,45,60,68,51,65,24,46)(13,31,52,63,36,58,59,61,34,57,67,66,53,62,35,39);(2,4)(5,8)(6,13)(7,19)(9,24)(10,25)(11,23)(12,31)(14,35)(15,36)(16,34)(17,39)(18,44)(20,46)(21,47)(22,45)(26,52)(27,51)(28,50)(29,53)(30,55)(32,58)(33,57)(37,60)(38,59)(40,62)(41,63)(42,61)(43,64)(48,65)(49,66)(54,67)(56,68);(8,23)(13,34)(19,45)(24,50)(25,51)(31,57)(35,59)(36,53)(39,61)(44,60)(46,55)(47,65)(52,67)(58,62)(63,66)(64,68)

Polycyclic presentation:

2^8|a^2=h,b^2=eg,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#141446, #141449, #141501, …
maximal normal subgroup of	#141446, #141449, #141501, …
maximal quotient of	#490475, #490546, #490577, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26596);