#7623365

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623308
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1652 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)(11,21)(12,14)(13,25)(16,29)(18,20)(19,34)(22,38)(23,37)(24,27)(26,28)(30,39)(31,40)(32,33)(35,36);(1,3,4,8)(2,5,6,7)(9,10,15,17)(11,16,23,30)(12,18,14,20)(13,19,27,32)(21,29,37,39)(22,31,28,36)(24,33,25,34)(26,35,38,40);(3,8)(5,7)(9,11)(10,16)(12,22)(13,26)(14,28)(15,23)(17,30)(18,31)(19,35)(20,36)(21,25)(24,37)(27,38)(29,34)(32,40)(33,39);(2,6)(5,7)(10,17)(16,30)(18,20)(19,32)(29,39)(31,36)(33,34)(35,40);(9,12,15,14)(10,18,17,20)(11,22,23,28)(13,24,27,25)(16,31,30,36)(19,33,32,34)(21,26,37,38)(29,35,39,40)

Polycyclic presentation:

2^9|b^2=fi,e^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	#72343243, #72343250, #72343273
maximal normal subgroup of	#72343243, #72343250, #72343273
maximal quotient of	#72343710

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530561);