#99669

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

849 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 849 conjugacy classes of subgroups — too many to draw. Only its 146 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)(5,6,10,11,9,7,8,12)(13,14)(15,19,20,17)(16,18)(21,22)(23,29,31,36)(24,26)(25,28)(27,30,34,33)(32,35);(1,2,3,4)(5,7,9,6)(8,12,10,11)(13,14)(15,17)(16,18)(19,20)(22,26)(25,32)(27,34)(30,33);(5,8,9,10)(6,11,7,12)(13,15,16,20)(14,17,18,19)(21,23)(22,27)(24,31)(25,33)(26,34)(28,36)(29,35)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6865);