#433296

order	512
nilpotency class	3
isoclinism	isoclinic class of #420416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15476
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1069 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:

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

Polycyclic presentation:

2^9|a^2=g,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,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340492);