#6986088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1164 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fh,c^2=fgi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6893284);