#8464378

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^2=hi,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, 8371574);