#130039

order	512
nilpotency class	4
isoclinism	isoclinic class of #124861
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1558
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

586 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 122 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)(5,6,8,14,22,15,32,31,10,12,21,28,9,13,23,29)(7,16,18,30,24,26,20,35,17,33,34,36,19,11,25,27);(5,7,10,17)(6,11)(8,20,21,25)(9,24,22,19)(12,26)(13,16)(14,30)(15,33)(18,23,34,32)(27,31)(28,36)(29,35);(2,4)(6,12)(8,21)(11,26)(13,15)(16,33)(18,34)(20,25)(23,32)

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=hi,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,f^b=fgh,g^b=gi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37235);