#8094843

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	#45482	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002039);