#254177

order	512
nilpotency class	3
isoclinism	isoclinic class of #254131
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	C₂³	C₂×#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

758 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,c^2=i,d^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,g^b=gi,d^c=di,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, 161373);