#7501000

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
prime radical	C₂×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3167 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)(9,10,14,18,15,13,26,28)(11,19,21,27,22,33,29,16)(12,23,25,39,24,17,31,32)(20,35,37,30,36,34,40,38);(5,6)(7,8)(9,11,15,22)(10,16,13,27)(12,20,24,36)(14,29,26,21)(17,30,23,38)(18,33,28,19)(25,40,31,37)(32,35,39,34);(2,4)(6,8)(9,12)(10,17)(11,20)(13,23)(14,25)(15,24)(16,30)(18,32)(19,34)(21,37)(22,36)(26,31)(27,38)(28,39)(29,40)(33,35);(2,4)(9,13,15,10)(11,19,22,33)(12,23,24,17)(14,18,26,28)(16,29,27,21)(20,34,36,35)(25,32,31,39)(30,40,38,37);(12,24)(17,23)(20,36)(25,31)(30,38)(32,39)(34,35)(37,40)

Polycyclic presentation:

2^9|a^2=f,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, 7408196);