#255088

order	512
nilpotency class	3
isoclinism	isoclinic class of #246503
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	#59041	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1244 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=ei,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 162284);