#7662439

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	#37277	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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,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, 7569635);