#8533401

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8440597);