#8227499

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

4051 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=hi,e^2=gi,f^2=g,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, 8134695);