#405922

order	512
nilpotency class	3
isoclinism	isoclinic class of #395587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14921
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1176 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)(3,15,18,4)(5,20,22,11)(6,12)(7,13)(8,14)(9,17,23,10)(16,19)(21,28,24,27)(25,30)(26,32)(29,31);(1,3,6,17,8,18,7,10)(2,9,13,4,14,23,12,15)(5,16,21,30,22,31,24,26)(11,25,28,19,20,32,27,29);(1,4,6,9,8,15,7,23)(2,10,12,3,14,17,13,18)(5,19,21,25,22,29,24,32)(11,26,27,16,20,30,28,31);(1,5)(2,11)(3,16)(4,19)(6,21)(7,24)(8,22)(9,25)(10,26)(12,27)(13,28)(14,20)(15,29)(17,30)(18,31)(23,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319325, #72319399, #72319444
maximal normal subgroup of	#72319325, #72319399, #72319444

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313118);