#8750884

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	#45046	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34,39,38)(35,44,46,40)(36,42,47,41)(37,48,45,43);(1,3,7,4)(2,8,11,14)(5,12,10,15)(6,13,9,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35)(34,40)(36,37)(38,44)(39,46)(41,48)(42,43)(45,47);(3,4)(8,14)(12,15)(13,16)(20,28)(25,30)(27,29)(31,32)(33,36,39,47)(34,41,38,42)(35,37,46,45)(40,48,44,43);(1,4,7,3)(2,8,11,14)(5,15,10,12)(6,13,9,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,36)(34,42)(35,45)(37,46)(38,41)(39,47)(40,48)(43,44);(1,5)(2,9)(3,12)(4,15)(6,11)(7,10)(8,16)(13,14)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32)(33,37,39,45)(34,43,38,48)(35,36,46,47)(40,42,44,41);(1,6,7,9)(2,10,11,5)(3,13,4,16)(8,15,14,12)(17,22,23,18)(19,26,21,24)(20,25,28,30)(27,31,29,32)(33,38,39,34)(35,44,46,40)(36,42,47,41)(37,43,45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,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, 8658080);