#420409

order	512
nilpotency class	3
isoclinism	isoclinic class of #420407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1708 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=ei,d^2=e,f^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,f^b=fi,h^b=hi,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, 327605);