#420386

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1399 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:

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

Polycyclic presentation:

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

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, 327582);