#8445894

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,12,14,9)(4,10,5,15)(6,16,13,11)(17,18,23,27)(19,28,29,24)(20,25,30,32)(21,22,26,31)(33,34,39,42)(35,43,46,38)(36,37,41,40)(44,47,48,45);(1,3)(2,9)(4,6)(5,13)(7,12)(8,14)(10,11)(15,16)(19,29)(20,30)(24,28)(25,32)(33,35,39,46)(34,38,42,43)(36,44,41,48)(37,45,40,47);(1,4,8,5)(2,10,7,15)(3,6,14,13)(9,11,12,16)(17,19)(18,24)(20,26)(21,30)(22,25)(23,29)(27,28)(31,32)(33,36,39,41)(34,40,42,37)(35,44,46,48)(38,47,43,45);(1,5,8,4)(2,10,7,15)(3,13,14,6)(9,11,12,16)(17,20,23,30)(18,25,27,32)(19,26,29,21)(22,28,31,24)(33,34)(35,43)(36,40)(37,41)(38,46)(39,42)(44,45)(47,48);(1,6,8,13)(2,11,7,16)(3,4,14,5)(9,10,12,15)(17,21,23,26)(18,22,27,31)(19,30,29,20)(24,25,28,32)(33,37,39,40)(34,41,42,36)(35,45,46,47)(38,48,43,44);(1,7,8,2)(3,12,14,9)(4,15,5,10)(6,16,13,11)(17,22)(18,26)(19,25)(20,24)(21,27)(23,31)(28,30)(29,32)(33,38,39,43)(34,35,42,46)(36,47,41,45)(37,48,40,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=gh,e^2=gi,f^2=g,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, 8353090);