#6482567

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,30,35)(27,37,40,32)(28,42,29,34)(31,46,45,36)(33,44,51,43)(38,48,39,52)(41,49,56,53)(47,54,50,55);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,30,40)(26,32,35,37)(28,39,29,38)(31,41,45,56)(33,50,51,47)(34,52,42,48)(36,49,46,53)(43,55,44,54);(11,17)(14,21)(18,19)(22,23)(25,28,30,29)(26,33,35,51)(27,38,40,39)(31,44,45,43)(32,47,37,50)(34,46,42,36)(41,55,56,54)(48,53,52,49);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,30,28)(26,34,35,42)(27,39,40,38)(31,44,45,43)(32,48,37,52)(33,46,51,36)(41,55,56,54)(47,53,50,49);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(26,35)(28,29)(32,37)(36,46)(38,39)(43,44)(49,53)(54,55)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6389763);