#423638

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1567 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:

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

Polycyclic presentation:

2^9|c^2=fi,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,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, 330834);