#411788

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35328	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=e,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318984);