#8425431

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

4234 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332627);