#331363

order	512
nilpotency class	3
isoclinism	isoclinic class of #331114
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2867	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 880 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)(3,6)(4,5)(7,8)(9,10,11,15,12,14,16,13)(17,18)(19,22)(20,23)(21,24)(25,29)(26,32)(27,31)(28,30);(2,5)(6,8)(17,19,25,26,20,27,21,28)(18,22,29,32,23,31,24,30);(3,7)(6,8)(10,13)(11,16)(14,15)(18,23)(19,26)(21,25)(22,30)(27,28)(31,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,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, 238559);