#8436641

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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)(8,11)(10,15)(17,18,24,22)(19,29,31,25)(20,27,21,26)(23,28,32,30)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,24,31)(18,25,22,29)(20,30,21,28)(23,27,32,26)(34,39)(35,42)(41,44)(43,48);(17,20,24,21)(18,26,22,27)(19,30,31,28)(23,29,32,25)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,21,24,20)(18,27,22,26)(19,30,31,28)(23,29,32,25)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,22,24,18)(19,29,31,25)(20,27,21,26)(23,30,32,28)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,23,24,32)(18,28,22,30)(19,27,31,26)(20,29,21,25)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=h,e^2=g,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, 8343837);