#138216

order	512
nilpotency class	4
isoclinism	isoclinic class of #125524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#2619	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

407 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 88 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)(5,6,9,16,11,7,13,19,12,17,27,25,10,15,26,31)(8,14,22,29,24,18,28,33,21,20,34,35,23,30,32,36);(5,7,9,25,11,6,13,19,12,15,27,16,10,17,26,31)(8,20,34,33,24,18,32,29,21,14,22,36,23,30,28,35);(2,4)(5,8,11,24,12,21,10,23)(6,14,7,18,17,20,15,30)(9,22,13,28,27,34,26,32)(16,29,19,33,25,35,31,36)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=h,d^2=gh,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,d^b=dh,f^b=fh,h^b=hi,d^c=di

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, 45412);