#8772807

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

4223 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:

(1,2,5,6)(3,9,11,7)(4,8,12,14)(10,16,13,15)(18,23)(20,24)(25,29)(28,31)(35,38)(40,43)(45,47)(46,48);(2,6)(4,12)(7,9)(10,13)(17,18,22,23)(19,24,27,20)(21,29,30,25)(26,28,32,31)(33,34,39,44)(35,43,38,40)(36,41,42,37)(45,46,47,48);(1,3,5,11)(2,7,6,9)(4,13,12,10)(8,16,14,15)(17,19,22,27)(18,24,23,20)(21,26,30,32)(25,31,29,28)(33,35,39,38)(34,40,44,43)(36,47,42,45)(37,48,41,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,13)(14,16)(17,20,22,24)(18,19,23,27)(21,28,30,31)(25,26,29,32)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,48,43,46);(1,4,5,12)(2,8,6,14)(3,10,11,13)(7,15,9,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,29)(30,32)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,47,43,45);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,21,22,30)(18,25,23,29)(19,26,27,32)(20,28,24,31)(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=gh,c^2=gi,d^2=gh,e^2=h,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, 8680003);