#92940

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₈	C₄×C₁₆
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1279
prime radical	C₂³	Q₈⋊C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

181 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 181 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fgh,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=ef,f^a=fh,c^b=ch

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136);