#383157

order	512
nilpotency class	3
isoclinism	isoclinic class of #213808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2032 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:

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

Polycyclic presentation:

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

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, 290353);