#8533089

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3884 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(17,18,22,26)(19,21,25,23)(20,28,30,24)(27,31,32,29)(34,36)(40,43)(41,42)(47,48);(2,7)(3,10)(8,14)(11,16)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,31,28,29)(33,34,39,36)(35,43,46,40)(37,41,38,42)(44,48,45,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,20,22,30)(18,24,26,28)(19,27,25,32)(21,29,23,31)(33,35,39,46)(34,40,36,43)(37,44,38,45)(41,47,42,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,29)(30,32)(33,36,39,34)(35,43,46,40)(37,42,38,41)(44,48,45,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28)(33,37,39,38)(34,41,36,42)(35,44,46,45)(40,47,43,48);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,32,28,27)(33,38,39,37)(34,42,36,41)(35,45,46,44)(40,48,43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440285);