#7623850

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623613
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	#61284	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1276 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531046);