#419848

order	512
nilpotency class	3
isoclinism	isoclinic class of #419450
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1676 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,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, 327044);