#10325398

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

4120 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=gh,e^2=g,f^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, 10232594);