#6720144

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

1397 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fi,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6627340);