#478196

order	512
nilpotency class	3
isoclinism	isoclinic class of #478195
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
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

960 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=ef,d^2=f,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,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385392);