#8464247

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	#37501	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8371443);