#135829

order	512
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#334
prime radical	C₂⁴	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

321 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,f^b=fi,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287284, D₄×#1369, #72319558, …
maximal normal subgroup of	#72287284, D₄×#1369, #72319558
maximal quotient of	#72286585

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43025);