#8324167

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

4126 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:

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

Polycyclic presentation:

2^9|a^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8231363);