#427245

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1346 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)(5,6,11,8)(7,12,17,19)(9,13,23,20)(10,21,26,14)(15,18,30,25)(16,28,22,27)(24,33,36,29)(31,35,34,32);(5,7)(6,12)(8,19)(9,22)(10,25)(11,17)(13,28)(14,30)(15,21)(16,23)(18,26)(20,27)(24,32)(29,31)(33,34)(35,36);(5,8,11,6)(7,15,17,30)(9,20,23,13)(10,14,26,21)(12,25,19,18)(16,31,22,34)(24,29,36,33)(27,35,28,32);(2,4)(5,9)(6,13)(7,16,17,22)(8,20)(10,24)(11,23)(12,27,19,28)(14,29)(15,31,30,34)(18,32,25,35)(21,33)(26,36)

Polycyclic presentation:

2^9|a^2=i,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, 334441);