#7500844

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3274 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)(9,10,16,33,15,20,14,19)(11,21,32,17,25,37,24,35)(12,26,27,31,18,13,28,30)(22,23,39,40,36,38,34,29);(5,6)(7,8)(9,11,15,25)(10,17,20,35)(12,22,18,36)(13,29,26,40)(14,32,16,24)(19,37,33,21)(23,30,38,31)(27,34,28,39);(2,4)(6,8)(9,12,15,18)(10,13,20,26)(11,22,25,36)(14,28,16,27)(17,29,35,40)(19,31,33,30)(21,38,37,23)(24,34,32,39);(2,4)(9,13)(10,18)(11,23)(12,20)(14,30)(15,26)(16,31)(17,34)(19,28)(21,22)(24,40)(25,38)(27,33)(29,32)(35,39)(36,37);(9,14,15,16)(10,19,20,33)(11,24,25,32)(12,27,18,28)(13,30,26,31)(17,21,35,37)(22,39,36,34)(23,40,38,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408040);