#8462124

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

4191 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)(3,13)(4,10)(5,7)(6,16)(8,12)(9,15)(11,14)(20,28)(25,29)(26,32)(30,31)(33,34,40,43)(35,44,39,41)(36,38,37,42)(45,47,46,48);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(17,18,22,21)(19,24,27,23)(20,25,28,29)(26,31,32,30)(33,35,40,39)(34,41,43,44)(36,46,37,45)(38,47,42,48);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,19)(18,23)(20,26)(21,24)(22,27)(25,30)(28,32)(29,31)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,47,44,48);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,32)(29,30)(33,37,40,36)(34,38,43,42)(35,46,39,45)(41,48,44,47);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,20,22,28)(18,25,21,29)(19,26,27,32)(23,30,24,31)(33,38)(34,36)(35,47)(37,43)(39,48)(40,42)(41,46)(44,45);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,21,22,18)(19,24,27,23)(20,29,28,25)(26,31,32,30)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,48,42,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=g,e^2=hi,f^2=ghi,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, 8369320);