#7656856

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	#34597	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4303 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564052);