#365045

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1634 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)(7,18)(8,22,26,33)(9,21)(10,16)(11,17)(12,14)(13,25,32,24)(15,27)(19,28,35,30)(20,29)(23,36,34,31);(1,2,4,3)(5,7,10,14,11,21,27,20)(6,12,15,18,17,29,16,9)(8,19,25,36,26,35,24,31)(13,30,33,34,32,28,22,23);(5,8)(6,13)(7,19)(9,23)(10,24)(11,26)(12,28)(14,31)(15,22)(16,33)(17,32)(18,34)(20,36)(21,35)(25,27)(29,30);(1,3,4,2)(5,9,10,12,11,18,27,29)(6,14,15,21,17,20,16,7)(8,23,24,28,26,34,25,30)(13,31,22,35,32,36,33,19)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=e,e^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,g^b=gi,f^c=fi,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, 272241);