#93139

order	512
nilpotency class	4
isoclinism	isoclinic class of #93139
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂⁵	C₄²
Frattini, prime radical, cosocle	C₂²×#127	C₂²
LCS	G ▷ C₂⁵ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂²×#127 ▷ C₂⁴ ▷ C₂² ▷ C₁

640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,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	#72234947, C₂×#72210016
maximal subgroup of	#72210015, #72210016, #72227527, …
maximal normal subgroup of	#72210015, #72210016, #72227527, …
maximal quotient of	#72210058, C₂×#93139, #72266345, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335);