#137001

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#447
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

370 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=h,d^2=gi,e^2=hi,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44197);