#7007824

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1353 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)(17,18,23,35,22,28,21,27)(19,24,36,46,31,40,30,39)(20,26,32,25,34,41,33,42)(29,38,43,37,45,47,44,48);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,31)(18,24,28,40)(20,29,34,45)(21,30,23,36)(25,37,42,48)(26,38,41,47)(27,39,35,46)(32,43,33,44);(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,29)(21,32)(22,34)(23,33)(24,37)(26,35)(27,41)(28,42)(30,43)(31,45)(36,44)(38,46)(39,47)(40,48);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(17,20)(18,26)(19,29)(21,33)(22,34)(23,32)(24,38)(25,35)(27,42)(28,41)(30,44)(31,45)(36,43)(37,46)(39,48)(40,47);(10,14)(13,15)(17,21,22,23)(18,27,28,35)(19,30,31,36)(20,32,34,33)(24,39,40,46)(25,41,42,26)(29,43,45,44)(37,47,48,38)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6915020);