#406348

order	512
nilpotency class	3
isoclinism	isoclinic class of #394464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1297 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)(9,10,13,18,15,19,14,29)(11,20,22,31,23,35,24,16)(12,17,27,34,26,33,28,40)(21,36,38,30,37,32,25,39);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,25)(13,24)(14,22)(15,23)(17,32)(18,35)(19,31)(20,29)(21,28)(26,38)(27,37)(30,34)(33,36)(39,40);(9,12)(10,17)(11,21,23,37)(13,27)(14,28)(15,26)(16,30,31,39)(18,34)(19,33)(20,32,35,36)(22,38,24,25)(29,40);(2,6)(5,8)(12,26)(17,33)(21,37)(25,38)(27,28)(30,39)(32,36)(34,40)

Polycyclic presentation:

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

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, 313544);