#8750866

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

3883 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18,23,22)(19,28,30,24)(20,26,31,25)(21,32,29,27)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,3,7,4)(2,8,11,14)(5,12,10,15)(6,13,9,16)(17,19)(18,24)(20,21)(22,28)(23,30)(25,32)(26,27)(29,31)(36,44)(39,46)(42,48)(45,47);(3,4)(8,14)(12,15)(13,16)(17,20,23,31)(18,25,22,26)(19,21,30,29)(24,32,28,27)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,4,7,3)(2,8,11,14)(5,15,10,12)(6,13,9,16)(17,20)(18,26)(19,29)(21,30)(22,25)(23,31)(24,32)(27,28)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,5)(2,9)(3,12)(4,15)(6,11)(7,10)(8,16)(13,14)(17,21,23,29)(18,27,22,32)(19,20,30,31)(24,26,28,25)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,6,7,9)(2,10,11,5)(3,13,4,16)(8,15,14,12)(17,22,23,18)(19,28,30,24)(20,26,31,25)(21,27,29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8658062);