#6487724

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470161
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1964 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:

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

Polycyclic presentation:

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

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, 6394920);