#359889

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2093 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)(7,12,19,29)(8,21,9,14)(10,16)(11,15)(13,22,32,23)(17,34)(18,27)(20,28,25,33)(24,31)(26,36)(30,35);(1,2)(3,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,25)(13,30)(14,27)(15,33)(16,29)(20,24)(21,34)(22,36)(23,35)(26,32)(28,31);(2,4)(5,8,10,9)(6,13)(7,17,19,18)(11,22,24,23)(12,26)(14,31)(15,21)(16,32)(20,35,25,36)(27,33)(28,34)(29,30);(5,9,10,8)(6,14,16,21)(7,18,19,17)(11,23,24,22)(12,27,29,34)(13,31,32,15)(20,36,25,35)(26,33,30,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267085);