#405233

order	512
nilpotency class	3
isoclinism	isoclinic class of #394040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35149	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2315 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,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, 312429);