#10328105

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

4090 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10235301);