#10102226

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

4101 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)(3,9)(4,5)(6,10)(7,13)(8,14)(11,15)(12,16)(18,22)(20,28)(23,25)(26,32)(33,34,39,44)(35,43,38,40)(36,46,47,41)(37,42,48,45);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,39,38)(34,40,44,43)(36,45,47,42)(37,46,48,41);(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,36,39,47)(34,41,44,46)(35,45,38,42)(37,43,48,40);(1,4,6,14)(2,8,10,5)(3,11,13,16)(7,15,9,12)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36)(34,41)(35,45)(37,40)(38,42)(39,47)(43,48)(44,46);(1,5,6,8)(2,4,10,14)(3,12,13,15)(7,11,9,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37)(34,42)(35,41)(36,43)(38,46)(39,48)(40,47)(44,45);(1,3)(2,9)(4,11)(5,15)(6,13)(7,10)(8,12)(14,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,38,39,35)(34,43,44,40)(36,42,47,45)(37,41,48,46)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10009422);