#454972

order	512
nilpotency class	3
isoclinism	isoclinic class of #454970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1516 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fi,h^c=hi,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, 362168);