#8696296

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

4090 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=gh,e^2=h,f^2=g,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, 8603492);