#8752046

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

4353 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)(3,9)(4,6)(5,8)(7,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(35,41)(39,42)(43,47)(46,48);(2,4)(5,13)(7,9)(11,16)(18,23)(21,24)(25,28)(29,31)(33,34)(35,42)(36,44)(37,38)(39,41)(40,45)(43,46)(47,48);(3,10)(7,9)(11,16)(14,15)(18,23)(19,26)(25,28)(27,32)(34,38)(39,42)(40,44)(46,48);(1,3,6,10)(2,7,4,9)(5,11,13,16)(8,14,12,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8659242);