#7501006

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

3231 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)(9,10,16,28,15,13,14,19)(11,20,29,17,23,33,22,27)(12,18,30,38,25,24,26,32)(21,34,39,31,35,40,36,37);(5,6)(7,8)(9,11,15,23)(10,17,13,27)(12,21,25,35)(14,29,16,22)(18,31,24,37)(19,33,28,20)(26,39,30,36)(32,40,38,34);(2,4)(6,8)(9,12)(10,18)(11,21)(13,24)(14,26)(15,25)(16,30)(17,31)(19,32)(20,34)(22,36)(23,35)(27,37)(28,38)(29,39)(33,40);(2,4)(9,13,15,10)(11,20,23,33)(12,24,25,18)(14,19,16,28)(17,29,27,22)(21,34,35,40)(26,32,30,38)(31,39,37,36);(12,25)(18,24)(21,35)(26,30)(31,37)(32,38)(34,40)(36,39)

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,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, 7408202);