#418320

order	512
nilpotency class	3
isoclinism	isoclinic class of #417601
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

1728 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)(8,22)(9,15)(10,17)(11,16)(13,23)(14,24)(18,27)(19,35)(20,32)(21,29)(25,28)(26,31)(30,33)(34,36);(1,2)(3,4)(5,7)(6,12)(8,19)(9,21)(10,25)(11,20)(13,30)(14,27)(15,29)(16,32)(17,28)(18,24)(22,35)(23,33)(26,34)(31,36);(5,8)(6,13)(7,18)(9,23)(10,26)(11,24)(12,27)(14,17)(15,22)(16,31)(19,25)(20,33)(21,34)(28,35)(29,36)(30,32);(1,3,4,2)(5,8)(6,14)(7,19)(9,23)(10,24)(11,26)(12,27)(13,17)(15,31)(16,22)(18,25)(20,34)(21,33)(28,30)(29,36)(32,35)

Polycyclic presentation:

2^9|d^2=e,g^2=i,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,g^b=gi,g^c=gi,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, 325516);