#419209

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3013	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1836 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)(7,18)(8,22)(9,14,27,33)(10,16)(11,15)(12,17)(13,20)(19,34,23,30)(21,29)(24,36,35,28)(25,32,26,31);(1,2)(3,4)(5,7,10,20,12,8,11,21)(6,13,15,18,17,29,16,22)(9,24,25,23,27,35,26,19)(14,30,31,36,33,34,32,28);(2,4)(5,8,12,7)(6,13,17,29)(9,23,27,19)(10,20,11,21)(14,28,33,36)(15,22,16,18)(24,26,35,25)(30,31,34,32);(5,9)(6,14)(7,19)(8,23)(10,26)(11,25)(12,27)(13,28)(15,32)(16,31)(17,33)(18,34)(20,35)(21,24)(22,30)(29,36)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,e^2=i,f^2=i,g^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,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi

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, 326405);