#7501002

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

3198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,4)(9,10,15,21,16,22,14,20)(11,23,25,34,12,18,27,17)(13,28,30,36,31,37,32,19)(24,33,29,35,26,40,39,38);(5,6)(7,8)(9,11)(10,17)(12,16)(13,29)(14,25)(15,27)(18,21)(19,35)(20,23)(22,34)(24,30)(26,32)(28,33)(31,39)(36,38)(37,40);(2,4)(6,8)(9,12,16,11)(10,18,22,23)(13,26,31,24)(14,25,15,27)(17,20,34,21)(19,35,36,38)(28,40,37,33)(29,30,39,32);(2,4)(9,13)(10,19)(11,24)(12,26)(14,30)(15,32)(16,31)(17,33)(18,35)(20,28)(21,37)(22,36)(23,38)(25,39)(27,29)(34,40);(9,14,16,15)(10,20,22,21)(11,25,12,27)(13,30,31,32)(17,23,34,18)(19,28,36,37)(24,39,26,29)(33,38,40,35)

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7408198);