#9057352

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

4142 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:

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

Polycyclic presentation:

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

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, 8964548);