#8425466

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

4214 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:

(1,2,6,10)(3,11,14,7)(4,15,13,8)(5,9,12,16)(17,18,23,21)(19,26,24,20)(22,25,31,29)(27,30,32,28)(33,34,39,45)(35,40,37,42)(36,44,47,41)(38,46,48,43);(1,3)(2,7)(4,12)(5,13)(6,14)(8,16)(9,15)(10,11)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,30)(31,32)(33,35)(34,40)(36,46)(37,39)(38,41)(42,45)(43,47)(44,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,16)(13,15)(17,18,23,21)(19,20,24,26)(22,25,31,29)(27,28,32,30)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,2)(3,11)(4,8)(5,16)(6,10)(7,14)(9,12)(13,15)(17,20,23,26)(18,24,21,19)(22,30,31,28)(25,27,29,32)(33,37,39,35)(34,42,45,40)(36,46,47,43)(38,44,48,41);(1,4,6,13)(2,8,10,15)(3,12,14,5)(7,16,11,9)(17,21,23,18)(19,26,24,20)(22,29,31,25)(27,30,32,28)(33,38)(34,43)(35,41)(36,42)(37,44)(39,48)(40,47)(45,46);(1,5,6,12)(2,9,10,16)(3,13,14,4)(7,15,11,8)(17,22,23,31)(18,25,21,29)(19,27,24,32)(20,28,26,30)(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=ghi,d^2=gh,e^2=g,f^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, 8332662);