#391968

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

843 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299164);