#8750895

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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)(17,18,22,21)(19,24,27,23)(20,28,29,25)(26,30,32,31)(33,34,39,42)(35,37,44,41)(36,45,38,40)(43,47,46,48);(1,3,7,4)(2,8,11,14)(5,12,10,15)(6,13,9,16)(17,19,22,27)(18,23,21,24)(20,26,29,32)(25,30,28,31)(33,35)(34,37)(36,46)(38,43)(39,44)(40,47)(41,42)(45,48);(3,4)(8,14)(12,15)(13,16)(17,19,22,27)(18,23,21,24)(20,26,29,32)(25,30,28,31)(33,36,39,38)(34,40,42,45)(35,43,44,46)(37,48,41,47);(1,4,7,3)(2,8,11,14)(5,15,10,12)(6,13,9,16)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,30)(29,32)(33,37,39,41)(34,35,42,44)(36,47,38,48)(40,46,45,43);(1,5)(2,9)(3,12)(4,15)(6,11)(7,10)(8,16)(13,14)(17,20,22,29)(18,25,21,28)(19,26,27,32)(23,30,24,31)(33,35)(34,41)(36,43)(37,42)(38,46)(39,44)(40,47)(45,48);(1,6,7,9)(2,10,11,5)(3,13,4,16)(8,15,14,12)(17,21,22,18)(19,24,27,23)(20,25,29,28)(26,30,32,31)(33,38,39,36)(34,40,42,45)(35,43,44,46)(37,47,41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,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, 8658091);