#395641

order	512
nilpotency class	3
isoclinism	isoclinic class of #394378
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2095 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,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, 302837);