#384774

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

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

829 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=f,e^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,h^b=hi,g^c=gi,h^c=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, 291970);