#461430

order	512
nilpotency class	3
isoclinism	isoclinic class of #454664
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)(5,6,11,17,12,16,10,31)(7,18,21,28,22,13,20,24)(8,19,27,33,25,36,26,14)(9,15,29,23,30,34,32,35);(5,7,12,22)(6,13,16,18)(8,23,25,35)(9,19,30,36)(10,21,11,20)(14,32,33,29)(15,26,34,27)(17,28,31,24);(5,8,10,26,12,25,11,27)(6,14,17,19,16,33,31,36)(7,15,20,35,22,34,21,23)(9,28,29,13,30,24,32,18);(2,4)(5,9)(6,15)(7,19)(8,24)(10,32)(11,29)(12,30)(13,26)(14,20)(16,34)(17,23)(18,27)(21,33)(22,36)(25,28)(31,35)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 368626);