#8755057

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

3983 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)(3,8)(4,6)(5,14)(7,9)(10,13)(11,16)(12,15)(17,18,24,29)(19,27,21,25)(20,26,22,31)(23,28,32,30)(34,38)(36,43)(39,44)(42,47);(1,3)(2,8)(4,13)(5,11)(6,10)(7,12)(9,15)(14,16)(17,19,24,21)(18,25,29,27)(20,30,22,28)(23,26,32,31)(35,41)(39,44)(42,47)(46,48);(2,9)(3,12)(4,11)(10,14)(17,20,24,22)(18,26,29,31)(19,30,21,28)(23,25,32,27)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4,7,11)(2,6,9,16)(3,13,12,5)(8,10,15,14)(17,21,24,19)(18,27,29,25)(20,30,22,28)(23,26,32,31);(1,5,7,13)(2,10,9,14)(3,11,12,4)(6,15,16,8)(17,22,24,20)(18,26,29,31)(19,30,21,28)(23,27,32,25)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,6)(2,11)(3,14)(4,9)(5,15)(7,16)(8,13)(10,12)(17,23,24,32)(18,28,29,30)(19,31,21,26)(20,25,22,27)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662253);