#6465508

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442584
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	#60415	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2511 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6372704);