#94287

order	512
nilpotency class	5
isoclinism	isoclinic class of #7410
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1344
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×#408	C₂²
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂×#408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

360 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 62 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 — 2 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,f^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,c^b=cg,d^b=di,g^b=gh,h^b=hi,e^c=eh,g^e=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1483);