#108567

order	512
nilpotency class	3
isoclinism	isoclinic class of #108562
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1434
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

611 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,f^b=fi,e^c=eh

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, 15763);