#8505446

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412642);