#513165

order	512
nilpotency class	5
isoclinism	isoclinic class of D₄×D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

455 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 122 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)(7,17,22,49)(8,23,27,59)(9,28,31,47)(10,32,30,15)(12,34,37,19)(13,38,40,25)(14,41,45,24)(18,39,52,66)(20,54,55,68)(21,56,33,48)(26,62,60,58)(29,42,64,67)(35,57,65,51)(36,53,46,50)(43,63,44,61);(5,7)(6,12)(8,18)(9,29)(10,33)(11,22)(13,39)(14,42)(15,46)(16,37)(17,23)(19,24)(20,25)(21,30)(26,57)(27,52)(28,35)(31,64)(32,36)(34,41)(38,55)(40,66)(43,54)(44,68)(45,67)(47,65)(48,62)(49,59)(50,61)(51,60)(53,63)(56,58);(2,4);(6,13)(7,18)(8,24)(9,25)(10,30)(12,17)(14,43)(15,47)(16,40)(19,20)(21,57)(22,52)(26,63)(27,41)(28,32)(29,50)(31,38)(33,51)(34,55)(35,54)(36,56)(37,49)(39,42)(44,45)(46,48)(53,64)(58,62)(60,61)(65,68)(66,67)

Polycyclic presentation:

2^9|a^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,g^a=gh,h^a=hi,d^b=de,e^b=eg,g^b=gh,h^b=hi,e^d=eg,g^d=gh,h^d=hi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234374, #72322380, #72322427
maximal normal subgroup of	#72234374, #72322380, #72322427

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420361);