#9018764

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3857 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8925960);