#7633945

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
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	#34976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4444 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)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(10,15)(14,20)(16,22)(21,24)(27,33)(30,37)(34,39)(38,40);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,22)(18,23)(27,33)(30,37)(34,39)(38,40);(10,15)(14,20)(16,22)(21,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,e^a=ehi,c^b=ci,d^b=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, 7541141);