#432890

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15468
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

983 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319831, #72319962, #72320009, …
maximal normal subgroup of	#72319831, #72319962, #72320009, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340086);