#8634314

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gh,f^2=gi,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, 8541510);