#8334399

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

4052 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8241595);