#471172

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

685 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:

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

Polycyclic presentation:

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

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, 378368);