#423489

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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1550 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)(5,6,11,16)(7,12,19,27)(8,21,23,30)(9,15,10,14)(13,24,29,22)(17,28,20,25)(18,32,33,36)(26,34,35,31);(1,2)(3,4)(5,7)(6,12)(8,18)(9,17)(10,20)(11,19)(13,26)(14,25)(15,28)(16,27)(21,32)(22,31)(23,33)(24,34)(29,35)(30,36);(7,17,19,20)(12,25,27,28)(18,31,33,34)(26,32,35,36);(1,3,4,2)(5,8)(6,13)(7,18)(9,22)(10,24)(11,23)(12,26)(14,21)(15,30)(16,29)(17,31)(19,33)(20,34)(25,32)(27,35)(28,36)

Polycyclic presentation:

2^9|a^2=i,c^2=f,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, 330685);