#8467225

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=h,e^2=ghi,f^2=gi,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, 8374421);