#383902

order	512
nilpotency class	3
isoclinism	isoclinic class of #30434
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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1578 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 291098);