#460568

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

904 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,h^c=hi,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, 367764);