#412539

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	#2537
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

974 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^c=ei,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, 319735);