#349932

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

985 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 985 conjugacy classes of subgroups — too many to draw. Only its 148 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,d^b=dh,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257128);