#8461307

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

4092 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,e^2=ghi,f^2=i,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, 8368503);