#461343

order	512
nilpotency class	3
isoclinism	isoclinic class of #455393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16387
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

614 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)(3,9,16,29)(4,18,20,27)(5,22)(6,14)(7,12)(8,13)(10,21,30,25)(11,15)(17,28,24,32)(19,26)(23,31);(1,3,7,17,6,16,8,24)(2,9,12,28,14,29,13,32)(4,19,21,15,20,23,25,5)(10,22,18,26,30,11,27,31);(1,4,6,20)(2,10)(3,15,16,5)(7,21,8,25)(9,26)(11,28)(12,18)(13,27)(14,30)(17,23,24,19)(22,32)(29,31);(1,5,6,15)(2,11)(3,4,16,20)(7,19,8,23)(9,27)(10,28)(12,31)(13,26)(14,22)(17,21,24,25)(18,29)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368539);