#8463190

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

3886 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=h,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, 8370386);