#122373

order	512
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
socle	C₂²	#1459
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

313 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=gh,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=chi,e^b=ef,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29569);