#128515

order	512
nilpotency class	3
isoclinism	isoclinic class of #105580
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1752
prime radical	C₂³	C₂×#97
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

259 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=i,e^2=h,f^2=g,h^2=i,b^a=bd,f^a=fgi,c^b=ch,d^b=dg,e^b=ei,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, 35711);