#6542469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3562	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,29)(21,32)(26,40)(27,35)(28,36)(31,38)(33,37)(34,42)(39,43)(41,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,29)(18,27)(19,28)(21,34)(22,36)(23,25)(24,35)(30,44)(31,43)(32,41)(33,42)(37,39)(38,40);(6,9)(10,12)(13,16,19,33)(14,21,24,38)(15,26,28,42)(17,31,18,32)(20,34,35,40)(22,30,23,37)(25,39,36,44)(27,41,29,43);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,28,29)(16,31,33,32)(20,25,35,36)(21,30,38,37)(26,41,42,43)(34,39,40,44);(7,11)(10,12)(15,28)(20,35)(25,36)(26,42)(27,29)(34,40)(39,44)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6449665);