#470587

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

874 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)(5,6,9,13,10,14,24,22)(7,15,19,16,18,11,25,27)(8,12,20,28,21,29,23,30)(17,31,35,33,34,26,36,32);(5,7,10,18)(6,11)(8,17,21,34)(9,19,24,25)(12,26)(13,27)(14,15)(16,22)(20,35,23,36)(28,32)(29,31)(30,33);(2,4)(5,6)(7,16)(8,12)(9,22)(10,14)(11,25)(13,24)(15,19)(17,33)(18,27)(20,30)(21,29)(23,28)(26,36)(31,35)(32,34);(5,8,10,21)(6,12,14,29)(7,17,18,34)(9,23,24,20)(11,26,15,31)(13,30,22,28)(16,32,27,33)(19,36,25,35)

Polycyclic presentation:

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

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, 377783);