#7045753

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45045	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6952949);