#8755294

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

3977 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662490);