#8429389

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

3945 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=gh,e^2=g,f^2=gi,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, 8336585);