#152783

order	512
nilpotency class	5
isoclinism	isoclinic class of #149927
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#535 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 558 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)(7,17,19,33)(8,16,10,13)(9,24,18,30)(11,15)(12,20,28,25)(14,26)(21,34,22,31)(23,27)(29,36,32,35);(5,7,18,35,11,21,9,25)(6,12,27,33,15,29,14,31)(8,19,23,36,10,22,26,20)(13,28,30,34,16,32,24,17);(2,4)(5,8,11,10)(6,13,15,16)(7,19,21,22)(9,26,18,23)(12,28,29,32)(14,24,27,30)(17,33,34,31)(20,35,36,25)

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,e^a=ei,g^a=gh,h^a=hi,d^b=dgh,e^b=ei,f^b=fghi,g^b=ghi,h^b=hi,d^c=di,f^c=fi,f^d=fhi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59979);