#470601

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

931 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)(5,6,10,16,11,14,9,15)(7,12,19,28,18,27,25,32)(8,20,23,13,24,31,22,30)(17,29,21,35,33,36,34,26);(5,7)(6,12,14,27)(8,21)(9,19)(10,25)(11,18)(13,29,30,36)(15,28,16,32)(17,23)(20,35,31,26)(22,33)(24,34);(2,4)(5,8,11,24)(6,13,14,30)(7,17,18,33)(9,23,10,22)(12,26,27,35)(15,31,16,20)(19,34,25,21)(28,36,32,29);(6,14)(9,10)(12,27)(17,33)(19,25)(20,31)(22,23)(26,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^c=gi,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, 377797);