#8445932

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	#54419	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4041 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=h,e^2=i,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, 8353128);