#152866

order	512
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

190 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=e,d^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dg,g^b=gh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60062);