#7658178

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	#38488	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4272 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=hi,e^2=h,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, 7565374);