#105547

order	512
nilpotency class	3
isoclinism	isoclinic class of #94966
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1062 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12743);