#427243

order	512
nilpotency class	3
isoclinism	isoclinic class of #417562
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1418 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)(9,22)(11,25)(12,19)(14,23)(17,24)(21,35)(29,30)(32,36);(5,6)(7,15)(8,19)(9,23)(10,13)(11,17)(12,20)(14,22)(16,30)(18,28)(21,27)(24,25)(26,36)(29,31)(32,33)(34,35);(5,7,10,18)(6,11,13,25)(8,16,20,31)(9,24,22,17)(12,26,19,33)(14,28,23,15)(21,36,35,32)(27,30,34,29);(2,4)(5,8)(6,12,13,19)(7,16)(9,21)(10,20)(11,26,25,33)(14,27,23,34)(15,29,28,30)(17,32)(18,31)(22,35)(24,36)

Polycyclic presentation:

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

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, 334439);