#457242

order	512
nilpotency class	3
isoclinism	isoclinic class of #457090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1328 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364438);