#455835

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,g^d=gi,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, 363031);