#7224433

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1255 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)(3,6)(4,8)(5,7)(11,15)(14,16)(17,18,22,28)(19,30,33,24)(20,26,36,45)(21,29,23,27)(25,38,43,35)(31,44,34,40)(32,41,37,46)(39,48,42,47);(1,3,6,2)(4,8,5,7)(9,10)(11,16)(12,13)(14,15)(17,19)(18,24)(20,34)(21,37)(22,33)(23,32)(25,42)(26,44)(27,46)(28,30)(29,41)(31,36)(35,47)(38,48)(39,43)(40,45);(1,4,6,5)(2,7,3,8)(10,13)(14,16)(17,20)(18,25)(19,31)(21,35)(22,36)(23,38)(24,39)(26,27)(28,43)(29,45)(30,42)(32,47)(33,34)(37,48)(40,46)(41,44);(1,4,6,5)(2,8,3,7)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,31)(21,38)(22,36)(23,35)(24,40)(25,27)(28,45)(29,43)(30,44)(32,48)(33,34)(37,47)(39,46)(41,42);(1,5,6,4)(2,8,3,7)(10,13)(14,16)(17,21,22,23)(18,27,28,29)(19,32,33,37)(20,35,36,38)(24,41,30,46)(25,26,43,45)(31,47,34,48)(39,44,42,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=fh,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,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, 7131629);