#7187283

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1216 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:

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

Polycyclic presentation:

2^9|b^2=fhi,c^2=fg,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7094479);