#383893

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,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, 291089);