#8741895

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

4145 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8649091);