#8463387

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=g,e^2=gi,f^2=hi,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, 8370583);