#7022910

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1362 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,29,30,41)(20,26,33,38)(21,28,23,27)(25,35,37,32)(31,43,44,48)(34,42,36,40)(39,45,47,46);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(20,33)(27,28)(32,35)(34,36)(39,47)(43,48);(9,11)(10,13)(12,16)(14,15)(17,19,21,35,22,30,23,32)(18,25,27,41,24,37,28,29)(20,31,36,45,33,44,34,46)(26,39,42,48,38,47,40,43);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(17,20)(18,26)(19,31)(21,36)(22,33)(23,34)(24,38)(25,39)(27,42)(28,40)(29,43)(30,44)(32,46)(35,45)(37,47)(41,48);(10,14)(13,15)(17,21,22,23)(18,27,24,28)(19,32,30,35)(20,34,33,36)(25,29,37,41)(26,40,38,42)(31,45,44,46)(39,48,47,43)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6930106);