#8750922

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	#36677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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)(17,18,23,25)(19,26,22,24)(20,21,30,31)(27,28,29,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3,7,4)(2,8,11,14)(5,12,10,15)(6,13,9,16)(17,19,23,22)(18,24,25,26)(20,29,30,27)(21,28,31,32)(34,38)(39,41)(40,45)(46,47);(3,4)(8,14)(12,15)(13,16)(18,25)(20,30)(24,26)(27,29)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4,7,3)(2,8,11,14)(5,15,10,12)(6,13,9,16)(17,20)(18,21)(19,27)(22,29)(23,30)(24,32)(25,31)(26,28)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5)(2,9)(3,12)(4,15)(6,11)(7,10)(8,16)(13,14)(17,21)(18,20)(19,28)(22,32)(23,31)(24,29)(25,30)(26,27)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47);(1,6,7,9)(2,10,11,5)(3,13,4,16)(8,15,14,12)(17,22,23,19)(18,24,25,26)(20,27,30,29)(21,28,31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=i,e^2=gh,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, 8658118);