#8096862

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

4101 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004058);