#6443049

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
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	#60444	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

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^c=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, 6350245);