#6465510

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442587
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	#60425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2479 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:

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

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^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, 6372706);