#8633931

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

4005 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8541127);