#384338

order	512
nilpotency class	3
isoclinism	isoclinic class of #219532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1059 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291534);