#457104

order	512
nilpotency class	3
isoclinism	isoclinic class of #457102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1140 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=i,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=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, 364300);