#8096813

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

4075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8004009);