#465215

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

1077 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)(5,6,11,15)(7,16,20,18)(8,13,23,31)(9,24,14,10)(12,27,29,19)(17,34,21,33)(22,32,26,25)(28,36,30,35);(5,7,10,19,11,20,24,27)(6,12,14,16,15,29,9,18)(8,21,26,35,23,17,22,36)(13,30,32,34,31,28,25,33);(5,8)(6,13)(7,17,20,21)(9,25)(10,26)(11,23)(12,28,29,30)(14,32)(15,31)(16,33,18,34)(19,36,27,35)(22,24);(2,4)(5,9,10,6,11,14,24,15)(7,18,19,12,20,16,27,29)(8,13,22,25,23,31,26,32)(17,28,35,34,21,30,36,33)

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=f,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=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, 372411);