#433919

order	512
nilpotency class	3
isoclinism	isoclinic class of #433919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15493
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1220 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341115);