#93144

order	512
nilpotency class	4
isoclinism	isoclinic class of #93139
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	C₂⁵	C₄²
Frattini, prime radical, cosocle	C₂²×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂²×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=h,e^2=hi,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	#72235029, #72235030, #72235536, …
Fitting subgroup of	#72235029
maximal subgroup of	#72235022, #72235029, #72235163, …
maximal normal subgroup of	#72235022, #72235029, #72235163, …
maximal quotient of	#72236026

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340);