#8772813

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

4241 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)(3,5)(4,9)(6,7)(8,11)(10,13)(12,16)(14,15)(17,18,23,27)(19,26,21,24)(20,22,31,25)(28,30,29,32)(35,38)(40,43)(45,47)(46,48);(2,7)(5,8)(9,15)(13,16)(17,19,23,21)(18,24,27,26)(20,28,31,29)(22,32,25,30)(33,34,39,44)(35,43,38,40)(36,41,42,37)(45,46,47,48);(1,3,6,11)(2,5,7,8)(4,12,14,10)(9,16,15,13)(17,20,23,31)(18,25,27,22)(19,28,21,29)(24,30,26,32)(33,35,39,38)(34,40,44,43)(36,47,42,45)(37,48,41,46);(1,3)(2,8)(4,10)(5,7)(6,11)(9,16)(12,14)(13,15)(17,21,23,19)(18,26,27,24)(20,28,31,29)(22,32,25,30)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,48,43,46);(1,4,6,14)(2,9,7,15)(3,10,11,12)(5,13,8,16)(17,20)(18,22)(19,29)(21,28)(23,31)(24,30)(25,27)(26,32)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,47,43,45);(1,5,6,8)(2,3,7,11)(4,13,14,16)(9,10,15,12)(17,22)(18,20)(19,30)(21,32)(23,25)(24,29)(26,28)(27,31)(33,38,39,35)(34,43,44,40)(36,47,42,45)(37,48,41,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=h,e^2=gh,f^2=gh,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, 8680009);