#10422715

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

4190 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)(3,4)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(18,19)(20,27)(21,29)(30,31)(33,34,39,43)(35,42,46,40)(36,38,37,41)(44,45,47,48);(1,2,6,10)(3,7,8,4)(5,9,14,16)(11,15,13,12)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,35)(34,40)(36,47)(37,44)(38,45)(39,46)(41,48)(42,43);(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,36,39,37)(34,41,43,38)(35,44,46,47)(40,45,42,48);(1,3)(2,7)(4,10)(5,13)(6,8)(9,12)(11,14)(15,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,37,39,36)(34,38,43,41)(35,44,46,47)(40,45,42,48);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,13,16,11)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,35)(34,42)(36,44)(37,47)(38,45)(39,46)(40,43)(41,48);(1,5,6,14)(2,9,10,16)(3,11,8,13)(4,12,7,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,36)(35,45)(37,43)(39,41)(40,47)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10329911);