#8230692

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

4047 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)(3,10)(4,14)(5,9)(6,7)(8,11)(12,13)(15,16)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(2,7)(3,11)(4,9)(13,16)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28)(33,35,39,36)(34,40,38,41)(37,44,46,43)(42,48,45,47);(2,7)(5,14)(8,10)(13,16)(18,23)(21,30)(24,27)(28,32)(33,35,39,36)(34,41,38,40)(37,43,46,44)(42,48,45,47);(1,3)(2,8)(4,12)(5,13)(6,11)(7,10)(9,15)(14,16)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32)(33,36,39,35)(34,41,38,40)(37,44,46,43)(42,48,45,47);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=ghi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=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, 8137888);