#6718664

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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)(12,13)(15,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,45)(30,38)(31,48)(32,41)(33,42)(34,43)(35,39)(36,44)(37,47)(40,46);(1,3,5,4)(2,6,7,8)(10,13)(11,15)(17,19)(18,23)(20,31)(21,29)(22,35)(24,40)(25,27)(26,38)(28,33)(30,34)(32,46)(36,47)(37,42)(39,43)(41,48)(44,45);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,28)(21,32)(22,36)(23,37)(25,41)(26,44)(27,45)(29,47)(30,46)(31,35)(33,34)(38,40)(39,48)(42,43);(10,13)(14,16)(19,29)(20,32)(23,27)(24,41)(30,35)(33,36)(38,39)(42,44);(1,4,5,3)(2,8,7,6)(18,25)(19,30)(22,34)(23,38)(24,41)(27,39)(28,46)(29,35)(31,47)(33,36)(37,40)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,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, 6625860);