#404082

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

1363 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)(7,12)(8,21)(9,17)(10,15)(11,16)(13,22)(14,19)(18,30)(20,28)(23,31)(24,32)(25,27)(26,33)(29,35)(34,36);(1,2,3,4)(5,7)(6,12)(8,18)(9,25)(10,19)(11,20)(13,29)(14,15)(16,28)(17,27)(21,30)(22,35)(23,33)(24,34)(26,31)(32,36);(5,8,11,24)(6,13,16,31)(7,18,20,34)(9,23,10,22)(12,26,28,29)(14,30,27,36)(15,32,17,21)(19,35,25,33);(5,7)(6,14,16,27)(8,18)(9,25)(10,19)(11,20)(12,15,28,17)(13,30,31,36)(21,26,32,29)(22,35)(23,33)(24,34)

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=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, 311278);