#434064

order	512
nilpotency class	3
isoclinism	isoclinic class of #419852
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 341260);