#427956

order	512
nilpotency class	3
isoclinism	isoclinic class of #418458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1255 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335152);