#422236

order	512
nilpotency class	3
isoclinism	isoclinic class of #421447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2474
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15303
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 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)(3,13,17,26)(4,18,21,19)(5,7)(6,11)(8,12)(9,15,24,16)(10,20,25,22)(14,23)(27,28)(29,31)(30,32);(1,3,7,15,8,17,6,16)(2,9,5,13,12,24,11,26)(4,14,20,29,21,30,22,28)(10,23,18,31,25,32,19,27);(1,4)(2,10)(3,14)(5,19)(6,20)(7,22)(8,21)(9,23)(11,18)(12,25)(13,27)(15,28)(16,29)(17,30)(24,32)(26,31);(1,5,6,2,8,11,7,12)(3,9,15,13,17,24,16,26)(4,19,20,10,21,18,22,25)(14,23,28,27,30,32,29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329432);