#8461200

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

4205 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,e^2=ghi,f^2=i,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, 8368396);