#8464116

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4010 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)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18,24,23)(19,28,30,25)(20,26,21,31)(22,32,29,27)(33,34)(35,40)(36,41)(37,47)(38,43)(39,44)(42,46)(45,48);(1,3,6,9)(2,7,10,5)(4,11,13,15)(8,14,16,12)(17,19)(18,25)(20,22)(21,29)(23,28)(24,30)(26,27)(31,32)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,48)(46,47);(1,3)(2,5)(4,11)(6,9)(7,10)(8,12)(13,15)(14,16)(17,20,24,21)(18,26,23,31)(19,22,30,29)(25,27,28,32)(33,35,39,48)(34,40,44,45)(36,46,43,37)(38,47,41,42);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,21,24,20)(18,26,23,31)(19,29,30,22)(25,27,28,32)(33,36)(34,41)(35,46)(37,48)(38,44)(39,43)(40,42)(45,47);(1,4,6,13)(2,8,10,16)(3,11,9,15)(5,12,7,14)(17,22,24,29)(18,27,23,32)(19,20,30,21)(25,26,28,31)(33,37,39,46)(34,42,44,47)(35,36,48,43)(38,45,41,40);(1,5,6,7)(2,9,10,3)(4,12,13,14)(8,15,16,11)(17,23,24,18)(19,28,30,25)(20,31,21,26)(22,32,29,27)(33,38,39,41)(34,43,44,36)(35,47,48,42)(37,45,46,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=g,e^2=gi,f^2=gi,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, 8371312);