#419455

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1354 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,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, 326651);