#128229

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1744
prime radical	C₂³	#431
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

260 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=dh,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fg,e^b=eg

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, 35425);