#254136

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	#36498	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 916 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) = 64

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=hi,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, 161332);