#8456964

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

4090 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=h,e^2=hi,f^2=g,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, 8364160);