#8437794

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=g,e^2=h,f^2=h,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, 8344990);