#10409680

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

4100 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)(3,11)(4,8)(5,6)(7,10)(9,16)(12,15)(13,14)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34,38,41)(35,42,36,39)(37,45,46,40)(43,47,44,48);(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3,7,4)(2,8,10,11)(5,12,9,14)(6,13,16,15)(17,19,22,20)(18,24,26,23)(21,27,29,28)(25,31,32,30)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,4,7,3)(2,8,10,11)(5,12,9,14)(6,15,16,13)(17,20,22,19)(18,24,26,23)(21,27,29,28)(25,30,32,31)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,5,7,9)(2,6,10,16)(3,12,4,14)(8,13,11,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,6)(2,9)(3,13)(4,15)(5,10)(7,16)(8,14)(11,12)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,37,38,46)(34,40,41,45)(35,43,36,44)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,e^2=h,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, 10316876);