#239022

order	512
nilpotency class	3
isoclinism	isoclinic class of #239017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60968	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

985 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 985 conjugacy classes of subgroups — too many to draw. Only its 190 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)(3,13)(4,8)(5,11)(6,10)(7,12)(9,15)(14,16)(17,18)(19,27)(20,23)(21,26)(22,25)(24,30)(28,31)(29,32);(1,3)(2,8)(4,5)(6,15)(7,14)(9,12)(10,13)(11,16)(18,23)(19,28)(20,21)(24,27)(25,26)(31,32);(1,4,6,14,7,15,5,3)(2,9,11,16,12,8,10,13)(17,19,20,29,22,28,21,30)(18,24,26,31,25,32,23,27);(1,5,7,6)(2,10,12,11)(3,4,14,15)(8,13,9,16)(18,25)(19,28)(23,26)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146218);