#364153

order	512
nilpotency class	3
isoclinism	isoclinic class of #363904
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2297
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13960
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1375 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=eg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319325, #72319359, #72319471
maximal normal subgroup of	#72319325, #72319359, #72319471

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271349);