#8455972

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

4120 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8363168);