#93152

order	512
nilpotency class	4
isoclinism	isoclinic class of #93145
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator, prime radical	C₂⁵	C₄²
Frattini, cosocle	C₂²×#126	C₂²
LCS	G ▷ C₂⁵ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×#126 ▷ C₂⁴ ▷ C₂² ▷ C₁

323 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=i,e^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=gi,c^b=cg,d^b=df,f^b=fi,g^b=gi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348);