#7623356

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623305
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,20)(12,17)(13,25)(14,29)(15,18)(16,22)(19,27)(21,36)(23,37)(24,34)(26,31)(28,39)(30,35)(32,38)(33,40);(1,3,4,8)(2,5,6,7);(3,8)(5,7)(9,11)(10,16)(12,21)(13,26)(14,30)(15,23)(17,31)(18,32)(19,35)(20,25)(22,27)(24,38)(28,40)(29,33)(34,36)(37,39);(2,6)(5,7)(9,12,15,14)(10,17,18,29)(11,21,23,30)(13,24,28,27)(16,31,32,33)(19,25,34,39)(20,36,37,35)(22,26,38,40);(10,18)(16,32)(17,29)(19,34)(20,37)(25,39)(31,33)(35,36)

Polycyclic presentation:

2^9|b^2=f,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gh,h^a=hi,c^b=cf,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310139, #72310140, #72310148
maximal normal subgroup of	#72310139, #72310140, #72310148

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530552);