#8463328

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(34,38)(39,44)(40,45)(46,48);(1,3,6,5)(2,7,9,10)(4,11,14,13)(8,15,12,16)(18,23)(19,26)(25,28)(27,32)(35,41)(39,44)(42,47)(46,48);(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,e^2=i,f^2=ghi,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, 8370524);