#93142

order	512
nilpotency class	4
isoclinism	isoclinic class of #93139
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

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

307 conjugacy classes of subgroups.

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:

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

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=ghi,d^c=dh,e^c=ehi,e^d=ei

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72234948, #72234949, #72235862, …
Fitting subgroup of	#72234948
maximal subgroup of	#72234500, #72234948, #72235797, …
maximal normal subgroup of	#72234500, #72234948, #72235797, …
maximal quotient of	#72237329

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338);