#419453

order	512
nilpotency class	3
isoclinism	isoclinic class of #419450
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1335 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=e,d^2=i,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,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, 326649);