#8755296

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

4165 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:

(1,2,5,8)(3,9,11,6)(4,7,13,16)(10,15,12,14)(17,18,23,27)(19,24,30,32)(20,22,31,25)(21,29,28,26)(33,34)(35,42)(36,44)(37,40)(38,39)(41,46)(43,47)(45,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,23,30)(18,24,27,32)(20,28,31,21)(22,26,25,29)(34,38)(40,42)(41,44)(47,48);(1,2,5,8)(3,6,11,9)(4,7,13,16)(10,14,12,15)(17,20,23,31)(18,25,27,22)(19,21,30,28)(24,26,32,29)(33,35,39,37)(34,40,38,42)(36,45,46,43)(41,48,44,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,28)(21,30)(23,31)(24,26)(25,27)(29,32)(33,36,39,46)(34,41,38,44)(35,43,37,45)(40,47,42,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,13)(15,16)(17,21,23,28)(18,26,27,29)(19,20,30,31)(22,24,25,32)(33,37,39,35)(34,42,38,40)(36,45,46,43)(41,48,44,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,22)(18,20)(19,29)(21,24)(23,25)(26,30)(27,31)(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=gi,d^2=hi,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, 8662492);