#471109

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16755
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

529 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)(3,12)(4,17,18,8)(5,11)(6,10)(7,15)(9,19,26,29)(13,22,14,25)(16,28)(20,21)(23,30,27,31)(24,32);(1,3,5,16,6,15,20,32)(2,7,21,28,10,12,11,24)(4,14,19,31,18,13,29,30)(8,25,26,27,17,22,9,23);(1,4)(2,8)(3,13,15,14)(5,19)(6,18)(7,22,12,25)(9,11)(10,17)(16,30,32,31)(20,29)(21,26)(23,24,27,28);(1,4)(2,9,10,26)(3,14)(5,19)(6,18)(7,23,12,27)(8,11,17,21)(13,15)(16,31)(20,29)(22,28,25,24)(30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=e,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,d^c=de,g^c=gi,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, 378305);