#8457063

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	#53702	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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)(3,10)(4,9)(5,14)(6,8)(7,11)(12,16)(13,15)(18,24)(19,20)(26,30)(27,29)(33,34,39,44)(35,40,46,48)(36,38,43,41)(37,42,45,47);(1,3,7,6)(2,8,11,10)(4,12,5,13)(9,15,14,16)(17,18,23,24)(19,25,20,28)(21,26,31,30)(22,27,32,29)(33,35,39,46)(34,40,44,48)(36,37,43,45)(38,42,41,47);(1,4,7,5)(2,9,11,14)(3,12,6,13)(8,15,10,16)(17,19,23,20)(18,25,24,28)(21,27,31,29)(22,26,32,30)(33,36,39,43)(34,41,44,38)(35,45,46,37)(40,42,48,47);(1,5,7,4)(2,9,11,14)(3,13,6,12)(8,15,10,16)(17,20,23,19)(18,25,24,28)(21,29,31,27)(22,26,32,30)(33,35)(34,40)(36,45)(37,43)(38,47)(39,46)(41,42)(44,48);(1,6,7,3)(2,10,11,8)(4,13,5,12)(9,16,14,15)(17,21,23,31)(18,26,24,30)(19,27,20,29)(22,25,32,28)(33,37,39,45)(34,42,44,47)(35,43,46,36)(38,40,41,48);(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,22,23,32)(18,27,24,29)(19,26,20,30)(21,25,31,28)(33,38,39,41)(34,43,44,36)(35,42,46,47)(37,40,45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=g,e^2=gi,f^2=h,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, 8364259);