#8425438

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

4236 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332634);