#422905

order	512
nilpotency class	3
isoclinism	isoclinic class of #418492
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60644	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1918 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330101);