#8192591

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

4119 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8099787);