#7623310

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623308
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1626 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)(11,16)(12,18)(13,26)(14,30)(15,19)(17,22)(20,28)(21,31)(23,32)(24,34)(25,37)(27,33)(29,40)(35,38)(36,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,24)(13,22)(14,21)(15,23)(17,26)(18,34)(19,32)(20,33)(25,39)(27,28)(29,38)(30,31)(35,40)(36,37);(3,8)(5,7)(9,11)(10,17)(12,21)(13,27)(14,24)(15,23)(16,26)(18,33)(19,35)(20,34)(22,28)(25,38)(29,39)(30,36)(31,37)(32,40);(2,6)(5,7)(9,12,15,14)(10,18,19,30)(11,21,23,24)(13,25,29,28)(16,31,32,34)(17,33,35,36)(20,26,37,40)(22,27,38,39);(10,19)(16,32)(17,35)(18,30)(20,37)(26,40)(31,34)(33,36)

Polycyclic presentation:

2^9|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,d^b=di,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343241, #72343252, #72343273
maximal normal subgroup of	#72343241, #72343252, #72343273
maximal quotient of	#72343697

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530506);