#8434146

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

4230 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)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(20,28)(24,31)(27,32)(29,30)(33,34,38,42)(35,39,40,36)(37,41,46,45)(43,47,48,44);(2,7)(3,10)(9,12)(11,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,35,38,40)(34,39,42,36)(37,43,46,48)(41,47,45,44);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,44)(47,48);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,48,45,43);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,37,38,46)(34,41,42,45)(35,43,40,48)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8341342);