#8696308

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

4162 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=h,e^2=gh,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, 8603504);