#455960

order	512
nilpotency class	3
isoclinism	isoclinic class of #455826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1175 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:

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

Polycyclic presentation:

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

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, 363156);