#7011396

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	#30414	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,e^2=fi,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, 6918592);