#7503342

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3608 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,f^b=fi,f^d=fi

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, 7410538);