#6720081

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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)(19,27)(20,32)(21,25)(22,26)(23,30)(24,34)(28,38)(29,40)(31,39)(33,41)(35,44)(36,43)(37,42)(45,46)(47,48);(1,3,5,4)(2,6,7,8)(9,11,14,10)(12,16,13,15)(17,19,21,30)(18,23,25,27)(20,33,34,47)(22,29,36,31)(24,41,32,48)(26,39,43,40)(28,37,46,35)(38,44,45,42);(2,7)(6,8)(9,12,14,13)(10,15,11,16)(17,20)(18,24)(19,28)(21,34)(22,37)(23,38)(25,32)(26,44)(27,45)(29,33)(30,46)(31,47)(35,36)(39,41)(40,48)(42,43);(9,12,14,13)(10,15,11,16)(18,25)(20,34)(23,27)(26,43)(28,46)(33,47)(35,37)(39,40);(1,4,5,3)(2,8,7,6)(9,13,14,12)(10,15,11,16)(19,29)(22,36)(23,39)(26,43)(27,40)(28,33)(30,31)(35,37)(38,41)(42,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fi,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,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, 6627277);