#8096930

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004126);