#7503338

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^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, 7410534);