#10352431

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

4253 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)(4,8)(5,6)(7,10)(11,16)(12,14)(13,15)(17,18,22,20)(19,23,25,27)(21,24,30,28)(26,31,29,32)(33,34,40,43)(35,44,38,41)(36,39,37,42)(45,46,47,48);(2,6)(3,10)(8,14)(11,13)(17,19)(18,23)(20,27)(21,29)(22,25)(24,32)(26,30)(28,31)(33,35,40,38)(34,41,43,44)(36,47,37,45)(39,46,42,48);(4,12)(8,14)(11,13)(15,16)(19,25)(23,27)(26,29)(31,32)(33,36,40,37)(34,42,43,39)(35,45,38,47)(41,46,44,48);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,12)(14,15)(17,20,22,18)(19,23,25,27)(21,28,30,24)(26,31,29,32)(33,37,40,36)(34,39,43,42)(35,45,38,47)(41,46,44,48);(1,4,5,12)(2,8,6,14)(3,11,10,13)(7,15,9,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,38,40,35)(34,41,43,44)(36,47,37,45)(39,48,42,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,39)(34,36)(35,46)(37,43)(38,48)(40,42)(41,47)(44,45)

Polycyclic presentation:

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

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, 10259627);