#383903

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	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1546 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)(7,17)(8,22)(9,24)(10,26)(11,16)(12,20)(13,23)(14,25)(15,28)(18,33)(19,35)(21,30)(27,32)(29,36)(31,34);(5,7,10,27,11,20,28,21)(6,12,15,32,16,17,26,30)(8,19,23,34,9,18,25,36)(13,29,22,33,14,31,24,35);(5,8,11,9)(6,13,16,14)(7,18,20,19)(10,25,28,23)(12,29,17,31)(15,24,26,22)(21,36,27,34)(30,33,32,35);(5,9,11,8)(6,14,16,13)(7,19,20,18)(10,23,28,25)(12,29,17,31)(15,22,26,24)(21,34,27,36)(30,33,32,35)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=i,d^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, 291099);