#106152

order	512
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1210 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1210 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,f^b=fhi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13348);