#8505523

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

4039 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)(4,11)(5,8)(6,7)(10,16)(12,13)(14,15)(17,18,23,27)(19,28,20,24)(21,25,31,29)(22,26,32,30)(33,34,39,36)(35,40,45,43)(37,46,48,41)(38,42,44,47);(2,7)(4,9)(8,15)(12,16)(19,20)(21,31)(24,28)(25,29)(33,35)(34,40)(36,43)(37,38)(39,45)(41,47)(42,46)(44,48);(17,19,23,20)(18,24,27,28)(21,26,31,30)(22,29,32,25)(33,34,39,36)(35,43,45,40)(37,41,48,46)(38,42,44,47);(1,3)(2,4)(5,13)(6,11)(7,9)(8,16)(10,14)(12,15)(17,20,23,19)(18,24,27,28)(21,30,31,26)(22,29,32,25)(33,36,39,34)(35,43,45,40)(37,46,48,41)(38,42,44,47);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,21,23,31)(18,25,27,29)(19,26,20,30)(22,28,32,24)(33,37)(34,41)(35,44)(36,46)(38,45)(39,48)(40,42)(43,47);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,22)(18,26)(19,29)(20,25)(21,28)(23,32)(24,31)(27,30)(33,38,39,44)(34,42,36,47)(35,37,45,48)(40,46,43,41)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8412719);