#7634856

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	#37277	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4391 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)(9,10,14,18)(11,15,13,17)(12,20,22,16)(19,24,21,23)(25,26,29,31)(27,30,33,38)(28,35,36,32)(34,40,39,37);(2,5)(6,8)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(26,31)(27,33)(32,35)(34,39);(9,11,14,13)(10,17,18,15)(12,21,22,19)(16,23,20,24)(25,28,29,36)(26,32,31,35)(27,34,33,39)(30,37,38,40);(9,13,14,11)(10,15,18,17)(12,21,22,19)(16,23,20,24);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,e^2=hi,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, 7542052);