#365067

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1650 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)(7,19)(8,14,26,32)(9,22)(10,17)(11,18)(12,16)(13,15)(20,24,34,30)(21,27)(23,36,35,29)(25,33,28,31);(1,2,4,3)(5,7,10,15,11,22,12,21)(6,13,16,19,18,27,17,9)(8,23,28,20,26,35,25,34)(14,30,33,29,32,24,31,36);(5,8)(6,14)(7,20)(9,24)(10,25)(11,26)(12,28)(13,29)(15,23)(16,31)(17,33)(18,32)(19,30)(21,35)(22,34)(27,36);(1,3,4,2)(5,9,10,13,11,19,12,27)(6,15,16,22,18,21,17,7)(8,24,25,29,26,30,28,36)(14,23,31,34,32,35,33,20)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^c=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, 272263);