#9815239

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

4174 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9722435);