#383894

order	512
nilpotency class	3
isoclinism	isoclinic class of #30441
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1531 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efi,e^2=i,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,e^b=ei,f^b=fi,f^c=fi,g^c=gi,e^d=ei,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, 291090);