#7623054

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622825
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
prime radical	#60458	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1636 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,15)(13,16)(14,26)(18,25)(19,24)(20,28)(21,30)(22,36)(23,27)(29,37)(31,35)(32,34)(33,38)(39,40);(2,5)(6,8)(10,15)(16,27)(17,28)(18,29)(19,26)(30,38)(31,39)(32,36);(1,3,4,7)(2,6,5,8)(10,16)(13,25)(14,24)(15,27)(17,30)(18,19)(21,35)(22,34)(23,37)(26,29)(28,38)(31,32)(33,40)(36,39);(9,11)(10,17)(12,20)(13,21)(14,22)(15,28)(16,30)(18,31)(19,32)(23,33)(24,34)(25,35)(26,36)(27,38)(29,39)(37,40);(11,20)(17,28)(21,33)(22,34)(30,38)(31,39)(32,36)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287046, #72320157, #72320496
maximal normal subgroup of	#72287046, #72320157, #72320496

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530250);