#8695326

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

4011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

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, 8602522);