#470279

order	512
nilpotency class	3
isoclinism	isoclinic class of #459488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
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

1221 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)(7,17)(8,22)(9,15)(10,16)(11,14)(12,20)(13,23)(18,27)(19,33)(21,28)(24,30)(25,29)(26,32)(31,35)(34,36);(5,7)(6,12,16,28)(8,19)(9,20)(10,18)(11,21)(13,26,29,33)(14,17,15,27)(22,31,30,36)(23,35)(24,32)(25,34);(2,4)(7,18)(8,23)(9,11)(13,22)(14,15)(17,27)(19,34)(24,25)(26,31)(29,30)(32,35)(33,36);(5,8,11,25,10,24,9,23)(6,13,14,22,16,29,15,30)(7,19,21,34,18,32,20,35)(12,26,17,31,28,33,27,36)

Polycyclic presentation:

2^9|b^2=gi,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,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, 377475);