#423477

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
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
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1637 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)(7,11)(8,12)(9,13)(10,14)(15,22)(16,30)(17,25)(18,24)(19,23)(20,29)(21,27)(26,32)(28,34)(31,36)(33,35);(1,2)(3,4)(5,7)(6,11)(8,19)(9,18)(10,17)(12,23)(13,24)(14,25)(15,26)(16,20)(21,33)(22,32)(27,35)(28,36)(29,30)(31,34);(5,8)(6,12)(7,16,17,19)(9,21)(10,20)(11,23,25,30)(13,27)(14,29)(15,28)(18,31,32,33)(22,34)(24,35,26,36);(1,3,4,2)(6,13)(9,22)(11,24)(12,27)(14,15)(18,32)(21,34)(23,35)(25,26)(28,29)(30,36)(31,33)

Polycyclic presentation:

2^9|c^2=f,d^2=e,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^c=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, 330673);