#108945

order	512
nilpotency class	3
isoclinism	isoclinic class of #95118
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 636 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,f^b=fh

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, 16141);