#8258113

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

4210 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8165309);