#10420072

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

4257 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=gh,f^2=g,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, 10327268);