#471210

order	512
nilpotency class	3
isoclinism	isoclinic class of #462392
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2517
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16778
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

517 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:

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

Polycyclic presentation:

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

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, 378406);