#403070

order	512
nilpotency class	3
isoclinism	isoclinic class of #396106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2459
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14866
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

833 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310266);