#418327

order	512
nilpotency class	3
isoclinism	isoclinic class of #417579
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
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1687 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,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, 325523);