#410409

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

779 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)(5,6,9,16,11,15,10,14)(7,17,19,29,20,12,26,28)(8,21,23,13,25,32,24,31)(18,34,22,33,35,30,36,27);(5,7,10,19,11,20,9,26)(6,12,14,28,15,17,16,29)(8,22,24,35,25,36,23,18)(13,30,21,27,31,34,32,33);(2,4)(5,8,9,23,11,25,10,24)(6,13,14,21,15,31,16,32)(7,18,19,22,20,35,26,36)(12,27,29,30,17,33,28,34);(6,14,15,16)(12,28,17,29)(13,21,31,32)(27,34,33,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317605);