#413123

order	512
nilpotency class	3
isoclinism	isoclinic class of #404632
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1018 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=e,d^2=h,e^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,e^b=ei,f^b=fi,f^c=fi

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, 320319);