#7180668

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1325 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)(12,20)(15,21)(19,24)(22,23)(25,26,32,38)(27,33,43,51)(28,44,29,35)(30,37,31,36)(34,46,52,45)(39,53,40,48)(41,50,42,49)(47,55,56,54);(2,5)(3,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,32,43)(26,33,38,51)(28,39,29,40)(30,41,31,42)(34,47,52,56)(35,48,44,53)(36,49,37,50)(45,54,46,55);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,28,32,29)(26,34,38,52)(27,39,43,40)(30,45,31,46)(33,47,51,56)(35,36,44,37)(41,54,42,55)(48,49,53,50);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(22,23)(25,29,32,28)(26,35,38,44)(27,40,43,39)(30,45,31,46)(33,48,51,53)(34,36,52,37)(41,54,42,55)(47,49,56,50);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(25,30,32,31)(26,36,38,37)(27,41,43,42)(28,45,29,46)(33,49,51,50)(34,44,52,35)(39,54,40,55)(47,53,56,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7087864);