#406354

order	512
nilpotency class	3
isoclinism	isoclinic class of #395552
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1286 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)(9,10,14,30,15,19,13,18)(11,20,29,16,23,32,22,33)(12,24,27,17,28,35,26,36)(21,31,25,34,39,40,38,37);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,25)(13,29)(14,22)(15,23)(17,34)(18,32)(19,33)(20,30)(21,27)(24,40)(26,39)(28,38)(31,35)(36,37);(9,12,14,27,15,28,13,26)(10,17,18,24,19,36,30,35)(11,21,22,38,23,39,29,25)(16,31,32,34,33,40,20,37);(2,6)(5,8)

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,g^b=gi,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, 313550);