#8445887

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=h,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, 8353083);