#465835

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

1050 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,e^d=ei,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, 373031);