#401798

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1188 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)(5,6,9,16,11,15,10,14)(7,17,19,29,20,12,25,28)(8,21,23,32,24,13,26,31)(18,27,22,30,35,33,34,36);(5,7)(6,12)(8,22)(9,25)(10,19)(11,20)(13,30)(14,28)(15,17)(16,29)(18,23)(21,36)(24,34)(26,35)(27,31)(32,33);(2,4)(5,8)(6,13)(7,18)(9,26)(10,23)(11,24)(12,27)(14,31)(15,21)(16,32)(17,33)(19,34)(20,35)(22,25)(28,36)(29,30);(6,14,15,16)(12,28,17,29)(13,31,21,32)(27,36,33,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308994);