#93188

order	512
nilpotency class	4
isoclinism	isoclinic class of #93181
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#125	C₂²
prime radical	C₂⁴	#95
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂²×#125 ▷ C₂⁴ ▷ C₂² ▷ C₁

291 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2)(3,11,13,25)(4,15,16,8)(5,9)(6,10)(7,12)(14,30,29,26)(17,23,18,22)(19,24)(20,31)(21,28)(27,32)(33,34,37,41,38,42,51,58)(35,43,45,59,46,60,50,63)(36,40,48,55,49,56,62,64)(39,52,53,44,54,61,57,47);(1,3,5,14,6,13,19,29)(2,7,9,20,10,21,24,32)(4,12,17,27,16,28,18,31)(8,11,22,26,15,25,23,30)(33,35)(34,39,42,54)(36,47,49,44)(37,50,51,45)(38,46)(40,43)(41,57)(48,61)(52,62)(53,58)(55,63,64,59)(56,60)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=i,e^2=h,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=gi,c^b=cg,d^b=dfh,f^b=fi,g^b=ghi,d^c=dh,e^c=ehi,e^d=ei

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384);