#94609

order	512
nilpotency class	6
isoclinism	isoclinic class of #7540
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1415
commutator	C₃₂	C₂×C₈
Frattini, cosocle	C₄×C₃₂	C₂²
prime radical	C₂×D₃₂	C₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2,3,5,4,6,7,8)(9,10,12,19,14,21,16,40)(11,23,26,50,28,17,30,41)(13,33,31,66,36,46,37,68)(15,39,32,53,38,69,35,22)(18,44,47,24,48,57,70,27)(20,52,51,25,45,60,67,34)(29,64,58,43,63,54,61,56)(42,71,49,62,55,65,72,59);(10,17)(11,24)(12,16)(13,34)(15,35)(18,45)(19,50)(20,48)(21,23)(22,54)(25,36)(26,44)(27,28)(29,65)(30,57)(31,60)(32,38)(33,42)(37,52)(39,43)(40,41)(46,55)(47,51)(49,68)(53,64)(56,69)(58,62)(59,61)(63,71)(66,72)(67,70)

Polycyclic presentation:

2^9|a^2=d,c^2=egi,d^2=f,e^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cei,d^b=di,e^b=eg,g^b=gh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230701
maximal normal subgroup of	#72230701
maximal quotient of	#72230775, #72230778

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1805);