#8493031

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

4265 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,e^2=hi,f^2=g,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, 8400227);