#416259

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	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

908 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)(5,6,12,17)(7,18)(8,24)(9,28,25,34)(10,16,11,30)(13,22)(14,26)(15,29,31,19)(20,35)(21,33)(23,32)(27,36);(5,7)(6,13,17,33)(8,25)(9,20)(10,21)(11,22)(12,23)(14,34,36,28)(15,24,31,35)(16,18,30,32)(19,26)(27,29);(2,4)(5,8)(6,14)(7,19)(9,21)(10,26)(11,27)(12,20)(13,31)(15,33)(16,35)(17,36)(18,34)(22,25)(23,29)(24,30)(28,32);(5,9,11,19,12,25,10,29)(6,15,16,28,17,31,30,34)(7,20,22,26,23,8,21,27)(13,24,18,14,33,35,32,36)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,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, 323455);