#8772788

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

4256 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,22,27)(19,26,30,23)(20,29,31,24)(21,25,32,28)(35,38)(40,43)(45,47)(46,48);(2,6)(4,13)(7,9)(11,16)(17,19,22,30)(18,23,27,26)(20,28,31,25)(21,29,32,24)(33,34,39,44)(35,43,38,40)(36,41,42,37)(45,46,47,48);(2,6)(3,10)(8,12)(11,16)(20,31)(21,32)(24,29)(25,28)(33,35,39,38)(34,40,44,43)(36,47,42,45)(37,48,41,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,20,22,31)(18,24,27,29)(19,28,30,25)(21,26,32,23)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,48,43,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,22,32)(18,25,27,28)(19,29,30,24)(20,26,31,23)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,47,43,45);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,19)(18,26)(20,28)(21,29)(22,30)(23,27)(24,32)(25,31)(33,38,39,35)(34,43,44,40)(36,47,42,45)(37,48,41,46)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8679984);