#7655062

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	#35102	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562258);