#8464172

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

4102 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)(3,7)(4,14)(5,9)(6,10)(8,11)(12,16)(13,15)(18,24)(19,20)(26,28)(27,30)(34,37)(36,40)(41,45)(44,46);(1,3,8,6)(2,7,11,10)(4,13,5,12)(9,16,14,15)(17,18)(19,25)(20,29)(21,26)(22,27)(23,24)(28,31)(30,32)(33,34,39,37)(35,40,43,36)(38,41,47,45)(42,46,48,44);(1,4,8,5)(2,9,11,14)(3,12,6,13)(7,15,10,16)(17,19,23,20)(18,25,24,29)(21,27,31,30)(22,28,32,26)(33,35,39,43)(34,40,37,36)(38,42,47,48)(41,46,45,44);(1,5,8,4)(2,9,11,14)(3,13,6,12)(7,15,10,16)(17,20,23,19)(18,25,24,29)(21,30,31,27)(22,28,32,26)(33,36)(34,35)(37,43)(38,46)(39,40)(41,48)(42,45)(44,47);(1,6,8,3)(2,10,11,7)(4,13,5,12)(9,16,14,15)(17,21,23,31)(18,26,24,28)(19,27,20,30)(22,29,32,25)(33,37,39,34)(35,36,43,40)(38,45,47,41)(42,44,48,46);(1,7,8,10)(2,3,11,6)(4,15,5,16)(9,12,14,13)(17,22)(18,27)(19,28)(20,26)(21,29)(23,32)(24,30)(25,31)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,46)(43,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=h,e^2=gi,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, 8371368);