#242389

order	512
nilpotency class	3
isoclinism	isoclinic class of #242389
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

791 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 791 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=dh,g^b=gi,e^c=eh,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149585);