#6443051

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60456	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350247);