#6482535

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

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,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, 6389731);