#8757295

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=gh,e^2=gi,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, 8664491);