#7305008

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,5)(3,7)(17,18,22,27)(19,23,29,39)(20,25,32,44)(21,34,36,26)(24,35,42,33)(28,38,31,43)(30,47,46,40)(37,48,41,45);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,29)(18,23,27,39)(20,31,32,28)(21,30,36,46)(24,41,42,37)(25,43,44,38)(26,40,34,47)(33,48,35,45);(9,11)(10,13)(12,16)(14,15)(17,20,22,32)(18,24,27,42)(19,28,29,31)(21,35,36,33)(23,37,39,41)(25,26,44,34)(30,48,46,45)(38,40,43,47);(2,5)(6,8)(10,14)(11,16)(17,20)(18,25)(19,28)(21,33)(22,32)(23,38)(24,34)(26,42)(27,44)(29,31)(30,45)(35,36)(37,47)(39,43)(40,41)(46,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(19,29)(20,32)(23,39)(24,42)(25,44)(30,46)(33,35)(40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7212204);