#456586

order	512
nilpotency class	3
isoclinism	isoclinic class of #456417
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1398 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 363782);