#7500830

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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

3225 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)(9,10,16,32,15,19,14,20)(11,21,30,17,25,37,24,36)(12,18,33,29,26,13,27,31)(22,23,35,34,38,40,39,28);(5,6)(7,8)(9,11,15,25)(10,17,19,36)(12,22,26,38)(13,28,18,34)(14,30,16,24)(20,37,32,21)(23,31,40,29)(27,35,33,39);(2,4)(6,8)(9,12)(10,18)(11,22)(13,19)(14,27)(15,26)(16,33)(17,34)(20,31)(21,23)(24,39)(25,38)(28,36)(29,32)(30,35)(37,40);(2,4)(9,13,15,18)(10,12,19,26)(11,23,25,40)(14,31,16,29)(17,35,36,39)(20,33,32,27)(21,38,37,22)(24,34,30,28);(10,19)(12,26)(17,36)(20,32)(21,37)(22,38)(27,33)(35,39)

Polycyclic presentation:

2^9|a^2=fi,b^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,e^c=ei,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, 7408026);