#253942

order	512
nilpotency class	3
isoclinism	isoclinic class of #253891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

773 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 773 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161138);