#8688199

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3877 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595395);