#7500994

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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3277 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)(9,10,15,30,14,19,12,17)(11,20,21,16,23,31,24,33)(13,26,25,18,29,34,28,36)(22,32,39,37,38,40,27,35);(5,6)(7,8)(9,11)(10,16)(12,21)(13,27)(14,23)(15,24)(17,31)(18,35)(19,33)(20,30)(22,28)(25,38)(26,32)(29,39)(34,40)(36,37);(2,4)(6,8)(9,12,14,15)(10,17,19,30)(11,21,23,24)(13,25,29,28)(16,31,33,20)(18,34,36,26)(22,27,38,39)(32,35,40,37);(2,4)(9,13,14,29)(10,18,19,36)(11,22,23,38)(12,25,15,28)(16,32,33,40)(17,34,30,26)(20,37,31,35)(21,27,24,39);(11,23)(16,33)(20,31)(21,24)(22,38)(27,39)(32,40)(35,37)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^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,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, 7408190);