#7228599

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200002
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1237 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=i,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7135795);