#6854967

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1449 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18,22,24)(19,28,29,42)(20,33,34,26)(21,27,23,37)(25,31,38,32)(30,45,44,43)(35,41,36,40)(39,46,48,47);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15)(18,24)(19,29)(20,34)(27,37)(31,32)(35,36)(39,48)(43,45);(9,10,13,12)(11,15,16,14)(17,19,22,29)(18,25,24,38)(20,30,34,44)(21,32,23,31)(26,39,33,48)(27,42,37,28)(35,47,36,46)(40,43,41,45);(2,6)(5,7)(17,20,22,34)(18,26,24,33)(19,30,29,44)(21,36,23,35)(25,39,38,48)(27,41,37,40)(28,43,42,45)(31,47,32,46);(3,8)(5,7)(9,12,13,10)(11,14,16,15)(17,21,22,23)(18,27,24,37)(19,31,29,32)(20,35,34,36)(25,28,38,42)(26,40,33,41)(30,46,44,47)(39,45,48,43)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6762163);