#9945589

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

4102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

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, 9852785);