#152945

order	512
nilpotency class	5
isoclinism	isoclinic class of #149923
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

338 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=e,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgh,f^b=fghi,g^b=ghi,h^b=hi,f^d=fhi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60141);