#7500859

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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3236 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)(9,10,13,19,14,18,30,36)(11,20,23,32,22,16,28,15)(12,24,26,17,27,35,29,34)(21,38,25,37,39,31,40,33);(5,6)(7,8)(9,11)(10,15)(12,25)(13,28)(14,22)(16,19)(17,33)(18,32)(20,36)(21,26)(23,30)(24,38)(27,40)(29,39)(31,35)(34,37);(2,4)(6,8)(9,11)(10,16)(12,21)(13,23)(14,22)(15,19)(17,33)(18,20)(24,31)(25,26)(27,39)(28,30)(29,40)(32,36)(34,37)(35,38);(2,4)(9,12,14,27)(10,17,18,34)(11,21,22,39)(13,29,30,26)(15,31,32,38)(16,33,20,37)(19,24,36,35)(23,40,28,25);(10,18)(11,22)(17,34)(19,36)(21,39)(23,28)(24,35)(25,40)

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,e^b=ei,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, 7408055);