#129718

order	512
nilpotency class	4
isoclinism	isoclinic class of #124651
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#3054	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

916 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 108 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)(5,6,8,13,11,15,10,26)(7,16,19,30,20,34,17,33)(9,22,24,14,25,31,21,32)(12,23,27,36,29,35,28,18);(5,7)(6,12)(8,17)(9,23)(10,19)(11,20)(13,27)(14,30)(15,29)(16,22)(18,24)(21,36)(25,35)(26,28)(31,34)(32,33);(2,4)(5,8,11,10)(6,13,15,26)(7,17,20,19)(9,21,25,24)(12,27,29,28)(14,22,32,31)(16,33,34,30)(18,23,36,35)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,h^b=hi,f^d=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, 36914);