#409172

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

764 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=f,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,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, 316368);