#404535

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1030 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)(5,6,9,15,11,16,10,14)(7,17,19,26,20,27,24,12)(8,13,22,29,21,30,23,31)(18,33,35,25,34,32,36,28);(5,7,9,24,11,20,10,19)(6,12,15,27,16,26,14,17)(8,18,22,36,21,34,23,35)(13,28,29,32,30,25,31,33);(5,8)(6,13)(7,18)(9,22)(10,23)(11,21)(12,25)(14,31)(15,29)(16,30)(17,32)(19,35)(20,34)(24,36)(26,28)(27,33);(2,4)(6,14,16,15)(8,21)(12,17,26,27)(13,29,30,31)(18,34)(22,23)(25,33,28,32)(35,36)

Polycyclic presentation:

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

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, 311731);