#8821808

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

4050 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(33,34,38,43)(35,42,46,39)(36,45,47,40)(37,41,48,44);(2,6)(4,13)(7,9)(11,16)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,35,38,46)(34,39,43,42)(36,44,47,41)(37,45,48,40);(2,6)(3,10)(8,12)(11,16)(18,20)(19,25)(24,28)(27,31)(36,47)(37,48)(40,45)(41,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,47)(34,40,43,45)(35,44,46,41)(37,42,48,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,37,38,48)(34,41,43,44)(35,45,46,40)(36,42,47,39);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,35)(34,42)(36,44)(37,45)(38,46)(39,43)(40,48)(41,47)

Polycyclic presentation:

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

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, 8729004);