#455963

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

1130 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,e^d=ei,f^d=fi,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, 363159);