#420410

order	512
nilpotency class	3
isoclinism	isoclinic class of #419792
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1678 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,g^c=gi,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, 327606);