#8692577

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

4018 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)(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(33,34,38,39)(35,40,36,41)(37,42,46,45)(43,47,44,48);(1,2,6,3)(4,7,11,9)(5,8,14,10)(12,15,13,16)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(34,39)(40,41)(42,45)(47,48);(1,2,6,3)(4,7,11,9)(5,8,14,10)(12,15,13,16)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,3,6,2)(4,9,11,7)(5,8,14,10)(12,15,13,16)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,48)(44,47);(1,4)(2,7)(3,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29)(33,36,38,35)(34,41,39,40)(37,43,46,44)(42,47,45,48);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=h,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, 8599773);