#8463186

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3891 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=h,e^2=hi,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, 8370382);