#8757061

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

3975 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=g,e^2=gi,f^2=gh,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, 8664257);