#7500854

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	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3204 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)(9,10,16,22,15,11,14,19)(12,20,24,34,26,17,25,33)(13,27,29,18,31,28,30,21)(23,36,40,35,39,37,38,32);(5,6)(7,8)(9,11,15,10)(12,17,26,20)(13,28,31,27)(14,19,16,22)(18,30,21,29)(23,37,39,36)(24,34,25,33)(32,40,35,38);(2,4)(6,8)(9,12,15,26)(10,17,11,20)(13,23,31,39)(14,25,16,24)(18,32,21,35)(19,34,22,33)(27,37,28,36)(29,40,30,38);(2,4)(9,13)(10,18)(11,21)(12,23)(14,29)(15,31)(16,30)(17,32)(19,28)(20,35)(22,27)(24,38)(25,40)(26,39)(33,37)(34,36);(9,14,15,16)(10,19,11,22)(12,24,26,25)(13,29,31,30)(17,33,20,34)(18,28,21,27)(23,38,39,40)(32,37,35,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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, 7408050);