#460562

order	512
nilpotency class	3
isoclinism	isoclinic class of #456658
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

958 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)(5,6,9,15,11,13,10,16)(7,12,20,25,19,27,17,30)(8,21,22,14,24,29,23,28)(18,32,31,36,34,35,33,26);(5,7)(6,12,13,27)(8,18)(9,17)(10,20)(11,19)(14,26,28,36)(15,30,16,25)(21,32,29,35)(22,33)(23,31)(24,34);(6,13)(7,17)(9,10)(12,25)(18,33)(19,20)(21,29)(22,23)(26,35)(27,30)(31,34)(32,36);(2,4)(5,8,11,24)(6,14,13,28)(7,18,19,34)(9,23,10,22)(12,26,27,36)(15,21,16,29)(17,31,20,33)(25,35,30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,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, 367758);