#7500852

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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×#3010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3210 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)(9,10,15,16,14,20,13,19)(11,17,22,29,24,33,23,32)(12,25,27,34,28,30,26,18)(21,35,38,40,37,39,36,31);(5,6)(7,8)(10,16)(12,26)(13,15)(17,29)(18,34)(19,20)(21,36)(22,23)(27,28)(31,40)(32,33)(37,38);(2,4)(6,8)(9,11,14,24)(10,17,20,33)(12,21,28,37)(13,23,15,22)(16,29,19,32)(18,31,34,40)(25,35,30,39)(26,36,27,38);(2,4)(9,12)(10,18)(11,21)(13,27)(14,28)(15,26)(16,30)(17,31)(19,25)(20,34)(22,36)(23,38)(24,37)(29,39)(32,35)(33,40);(9,13,14,15)(10,19,20,16)(11,22,24,23)(12,27,28,26)(17,29,33,32)(18,25,34,30)(21,36,37,38)(31,39,40,35)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408048);