#8755308

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

3973 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)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,23,27)(19,24,30,32)(20,22,31,25)(21,29,28,26)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(2,7)(5,8)(9,12)(13,15)(17,19,23,30)(18,24,27,32)(20,28,31,21)(22,26,25,29)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(2,7)(3,10)(9,12)(11,16)(17,20,23,31)(18,25,27,22)(19,21,30,28)(24,26,32,29)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,20)(18,22)(19,28)(21,30)(23,31)(24,26)(25,27)(29,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,21,23,28)(18,26,27,29)(19,20,30,31)(22,24,25,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,22)(18,20)(19,29)(21,24)(23,25)(26,30)(27,31)(28,32)(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=gh,c^2=i,d^2=hi,e^2=gh,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, 8662504);