#7641554

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4291 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi

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, 7548750);