#244407

order	512
nilpotency class	3
isoclinism	isoclinic class of #233521
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

844 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 178 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=efh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151603);