#7500827

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

3239 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)(9,10,13,19,14,18,27,15)(11,20,23,29,22,16,31,33)(12,24,25,17,26,30,28,34)(21,36,37,35,38,40,39,32);(5,6)(7,8)(10,15)(12,25)(13,27)(16,29)(17,34)(18,19)(20,33)(21,37)(23,31)(26,28)(32,35)(38,39);(2,4)(6,8)(9,11)(10,16)(12,21)(13,23)(14,22)(15,29)(17,32)(18,20)(19,33)(24,40)(25,37)(26,38)(27,31)(28,39)(30,36)(34,35);(2,4)(9,12,14,26)(10,17,18,34)(11,21,22,38)(13,28,27,25)(15,30,19,24)(16,32,20,35)(23,39,31,37)(29,36,33,40);(10,18)(11,22)(15,19)(17,34)(21,38)(23,31)(24,30)(37,39)

Polycyclic presentation:

2^9|a^2=f,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, 7408023);