#7196037

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1382 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gh,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

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, 7103233);