#251494

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 876 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) = 256

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

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

Polycyclic presentation:

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

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, 158690);