#9015630

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

4108 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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,15)(13,14)(17,18,22,23)(19,27,30,24)(20,25,31,29)(21,28,32,26)(33,34)(35,44)(36,47)(37,38)(39,40)(41,45)(42,46)(43,48);(1,3,7,6)(2,8,9,16)(4,12,14,11)(5,10,15,13)(18,23)(20,31)(24,27)(26,28)(34,40)(35,45)(37,43)(42,47);(2,9)(3,6)(4,14)(5,15)(17,19,22,30)(18,24,23,27)(20,26,31,28)(21,25,32,29)(33,35)(34,41)(36,43)(37,46)(38,47)(39,45)(40,44)(42,48);(1,4)(2,10)(3,12)(5,16)(6,11)(7,14)(8,15)(9,13)(17,19)(18,24)(20,26)(21,29)(22,30)(23,27)(25,32)(28,31)(33,36,39,46)(34,42,40,47)(35,43,45,37)(38,41,48,44);(1,5,7,15)(2,11,9,12)(3,13,6,10)(4,8,14,16)(17,20)(18,25)(19,28)(21,24)(22,31)(23,29)(26,30)(27,32)(33,37,39,43)(34,38,40,48)(35,46,45,36)(41,47,44,42);(1,6,7,3)(2,8,9,16)(4,11,14,12)(5,10,15,13)(17,21)(18,26)(19,29)(20,24)(22,32)(23,28)(25,30)(27,31)(33,38)(34,43)(35,47)(36,41)(37,40)(39,48)(42,45)(44,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=ghi,e^2=i,f^2=gh,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, 8922826);