#152767

order	512
nilpotency class	5
isoclinism	isoclinic class of #149923
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 531 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,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, 59963);