#8192472

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

4154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=h,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, 8099668);