#10102222

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

4209 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18,23,25)(19,26,22,24)(20,29,30,21)(27,32,31,28)(33,34)(35,42)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,23,22)(18,24,25,26)(20,27,30,31)(21,32,29,28)(35,37)(40,42)(43,44)(47,48);(2,4)(3,10)(9,12)(11,16)(18,25)(20,30)(24,26)(27,31)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20,23,30)(18,21,25,29)(19,27,22,31)(24,32,26,28)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21)(18,20)(19,28)(22,32)(23,29)(24,31)(25,30)(26,27)(33,37,39,35)(34,40,38,42)(36,43,46,44)(41,48,45,47);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,22,23,19)(18,26,25,24)(20,31,30,27)(21,28,29,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=h,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, 10009418);