#420429

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
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
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) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 327625);