#8772695

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

4274 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679891);