#465841

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1027 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,11,7,10,15,12,18)(8,19,22,29,21,16,26,13)(9,23,24,14,25,17,27,30)(20,33,34,31,36,28,35,32);(1,2,3,4)(5,7,11,15,10,18,12,6)(8,13,22,19,21,29,26,16)(9,17,24,30,25,23,27,14)(20,28,34,32,36,33,35,31);(2,4)(5,8,10,21)(6,13,15,29)(7,16,18,19)(9,20,25,36)(11,26,12,22)(14,28,30,33)(17,31,23,32)(24,35,27,34);(5,9,11,24,10,25,12,27)(6,14,7,17,15,30,18,23)(8,20,22,34,21,36,26,35)(13,28,19,32,29,33,16,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi,e^d=ei,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, 373037);