#412868

order	512
nilpotency class	3
isoclinism	isoclinic class of #402446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 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)(9,11)(13,27)(14,29)(15,24)(17,25)(18,32)(22,23)(30,33)(35,36);(1,2)(3,4)(5,6)(7,17,16,13)(8,14)(9,24)(10,15)(11,12)(18,31,30,28)(19,25,20,27)(21,26)(22,34)(23,29)(32,35,33,36);(2,4)(5,7,9,19,10,16,11,20)(6,13,12,25,15,17,24,27)(8,18,22,32,23,30,21,33)(14,28,26,35,29,31,34,36);(5,8,11,21,10,23,9,22)(6,14,12,26,15,29,24,34)(7,18,20,33,16,30,19,32)(13,28,25,35,17,31,27,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320064);