#8695913

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

4212 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=ghi,d^2=gh,e^2=g,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, 8603109);