#423488

order	512
nilpotency class	3
isoclinism	isoclinic class of #418504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1598 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:

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

Polycyclic presentation:

2^9|c^2=fi,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,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, 330684);