#390952

order	512
nilpotency class	3
isoclinism	isoclinic class of #32696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

779 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=gh,d^2=i,g^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,e^b=ei,h^b=hi,e^c=ei,f^c=fi,g^d=gi,h^d=hi

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, 298148);