#7199883

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1245 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=g,d^2=fhi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=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, 7107079);