#8425576

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

4083 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=g,e^2=h,f^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, 8332772);