#299576

order	512
nilpotency class	3
isoclinism	isoclinic class of #299541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

928 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206772);