#461426

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1416 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)(5,6,11,16,10,15,28,34)(7,17,20,35,21,12,29,30)(8,22,24,36,23,13,31,19)(9,25,26,18,27,14,32,33);(5,7)(6,12)(8,14)(9,19)(10,21)(11,29)(13,26)(15,17)(16,35)(18,24)(20,28)(22,32)(23,25)(27,36)(30,34)(31,33);(5,8,10,23)(6,13,15,22)(7,18)(9,12)(11,24,28,31)(14,20)(16,19,34,36)(17,27)(21,33)(25,29)(26,30)(32,35);(2,4)(5,9,10,27)(6,14,15,25)(7,19,21,36)(8,17,23,12)(11,26,28,32)(13,20,22,29)(16,33,34,18)(24,35,31,30)

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,g^b=gi,d^c=de,e^c=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, 368622);