#7634628

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#36865	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,19)(13,22)(14,17)(16,20)(18,24)(21,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(2,5)(6,8)(9,11,14,13)(10,15,17,22)(12,18,20,21)(16,23,19,24)(25,27,30,29)(26,31,34,33)(28,36,37,35)(32,39,40,38);(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,19)(22,24)(25,28,30,37)(26,32,34,40)(27,35,29,36)(31,38,33,39);(10,17)(12,20)(15,22)(18,21)(25,29,30,27)(26,33,34,31)(28,35,37,36)(32,38,40,39);(9,13,14,11)(10,15,17,22)(12,21,20,18)(16,23,19,24)(26,34)(31,33)(32,40)(38,39);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,e^a=ehi,c^b=ci,d^c=dhi

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, 7541824);