#434059

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
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	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1215 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)(5,6,10,16,9,15,11,17)(7,18,26,14,20,12,21,31)(8,22,24,30,25,32,27,13)(19,29,23,33,35,36,34,28);(5,7,9,20)(6,12,15,18)(8,23)(10,26,11,21)(13,29)(14,16,31,17)(19,27)(22,33)(24,35)(25,34)(28,32)(30,36);(5,8,11,24,9,25,10,27)(6,13,17,22,15,30,16,32)(7,19,26,34,20,35,21,23)(12,28,31,36,18,33,14,29);(2,4)(5,7)(6,14)(8,19)(9,20)(10,21)(11,26)(12,17)(13,29)(15,31)(16,18)(22,28)(23,27)(24,34)(25,35)(30,36)(32,33)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 341255);