#419913

order	512
nilpotency class	3
isoclinism	isoclinic class of #418492
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1558 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:

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

Polycyclic presentation:

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

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, 327109);