#433917

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	#23393	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1272 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)(5,6,9,14,11,16,10,15)(7,17,20,12,21,28,19,27)(8,13,24,30,25,32,23,31)(18,29,22,36,35,33,34,26);(5,7,9,20,11,21,10,19)(6,12,15,17,16,27,14,28)(8,22,23,18,25,34,24,35)(13,29,30,36,32,33,31,26);(5,8)(6,13)(7,18,21,35)(9,24)(10,23)(11,25)(12,26,27,36)(14,30)(15,31)(16,32)(17,33,28,29)(19,34,20,22);(2,4)(6,14)(7,19)(9,10)(12,27)(13,30)(15,16)(18,34)(20,21)(22,35)(23,24)(26,36)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^b=fi,e^c=ei,e^d=ei,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, 341113);