#403091

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,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, 310287);