#129723

order	512
nilpotency class	4
isoclinism	isoclinic class of #124940
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#3060	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 892 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)(7,17,19,12)(8,21,24,11)(9,15)(10,14)(13,22,26,35)(16,23)(18,25)(20,29,33,28)(27,32)(30,34)(31,36);(6,11)(8,22)(9,23)(12,25)(13,16)(14,21)(15,26)(17,30)(18,32)(20,33)(24,35)(27,29)(28,31)(34,36);(2,4)(5,7,10,19)(6,12,14,17)(8,18,24,34)(9,20,23,33)(11,25,21,30)(13,27,26,31)(15,28,16,29)(22,32,35,36)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,h^b=hi,f^c=fi,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, 36919);