#7634920

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4316 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=h,f^2=hi,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, 7542116);