#403008

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1222 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)(5,6,10,17,9,15,11,16)(7,12,20,28,21,29,19,13)(8,22,25,32,24,33,26,14)(18,34,36,31,23,30,35,27);(5,7,11,19,9,21,10,20)(6,12,17,28,15,29,16,13)(8,23,26,36,24,18,25,35)(14,31,22,30,32,27,33,34);(5,7)(6,13)(8,18)(9,21)(10,20)(11,19)(12,17)(14,30)(15,28)(16,29)(22,27)(23,24)(25,36)(26,35)(31,33)(32,34);(2,4)(5,8,11,26,9,24,10,25)(6,14,17,22,15,32,16,33)(7,18,19,35,21,23,20,36)(12,27,28,34,29,31,13,30)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

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, 310204);