#404044

order	512
nilpotency class	3
isoclinism	isoclinic class of #396101
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1359 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:

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

Polycyclic presentation:

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

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, 311240);