#8464384

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,e^2=ghi,f^2=gi,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, 8371580);