#8752526

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

4081 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=h,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, 8659722);