#331321

order	512
nilpotency class	3
isoclinism	isoclinic class of #331304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

755 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 755 conjugacy classes of subgroups — too many to draw. Only its 170 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)(9,10,14,15,13,11,12,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,30)(28,31)(29,32);(2,5)(6,8)(17,19,22,29,21,30,20,28)(18,23,26,31,24,27,25,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,10)(12,16,14,15)(18,24)(19,28)(20,22)(23,31)(27,32)(29,30);(3,7)(6,8)(9,12,13,14)(10,15,11,16)(17,20,21,22)(18,25,24,26)(19,29,30,28)(23,31,27,32)

Polycyclic presentation:

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

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, 238517);