#254250

order	512
nilpotency class	3
isoclinism	isoclinic class of #254227
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	#39500	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 851 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:

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

Polycyclic presentation:

2^9|b^2=ehi,c^2=i,d^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=deh,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161446);