#6502936

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6410132);