#253911

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	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 863 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) = 64

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

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

Polycyclic presentation:

2^9|b^2=ehi,c^2=i,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, 161107);