#405846

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
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	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

973 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)(7,17,20,14)(8,13)(9,15)(10,25)(11,16)(12,19,28,21)(18,29,34,33)(22,27,35,36)(23,30)(24,32)(26,31);(1,2)(3,4)(5,7)(6,12,16,28)(8,22,24,35)(9,19)(10,21)(11,20)(13,29)(14,15,17,25)(18,23,34,26)(27,30)(31,36)(32,33);(2,4)(5,8,9,23,11,24,10,26)(6,13,15,30,16,32,25,31)(7,18,19,22,20,34,21,35)(12,27,17,33,28,36,14,29);(5,7)(6,14)(8,18)(9,19)(10,21)(11,20)(12,15)(13,29)(16,17)(22,23)(24,34)(25,28)(26,35)(27,30)(31,36)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313042);