#7307945

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
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,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, 7215141);