#462447

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16432
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

881 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:

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

Polycyclic presentation:

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

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, 369643);