#412869

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×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1046 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)(8,21)(9,11)(13,25)(14,28)(15,24)(16,29)(17,31)(18,19)(22,23)(27,30)(32,33);(1,2)(3,4)(5,6,10,15)(7,16)(8,14,23,28)(9,24,11,12)(13,20)(17,30)(18,25)(19,29)(21,26,22,34)(27,33)(31,36)(32,35);(2,4)(5,7,9,18,10,20,11,19)(6,13,12,25,15,16,24,29)(8,17,22,32,23,33,21,31)(14,27,26,35,28,30,34,36);(5,8,9,22,10,23,11,21)(6,14,24,34,15,28,12,26)(7,17,18,32,20,33,19,31)(13,27,29,36,16,30,25,35)

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=hi,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, 320065);