#7187370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1191 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)(25,26,30,31)(27,36,39,49)(28,33,41,46)(29,34,44,35)(32,38,45,40)(37,51,52,56)(42,48,43,50)(47,54,55,53);(1,2,5,6)(3,7,4,8)(10,15)(11,18)(12,13)(23,24)(26,31)(28,41)(32,45)(34,35)(36,49)(37,52)(42,43)(53,54);(1,3,5,4)(2,7,6,8)(9,11,14,18)(10,16,15,22)(12,20,13,19)(17,24,21,23)(25,27,29,40,30,39,44,38)(26,32,35,36,31,45,34,49)(28,37,43,53,41,52,42,54)(33,47,48,51,46,55,50,56);(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)(29,43)(30,41)(31,46)(32,47)(34,50)(35,48)(36,51)(38,54)(39,52)(40,53)(42,44)(45,55)(49,56);(9,13,14,12)(10,17,15,21)(11,19,18,20)(16,24,22,23)(25,29,30,44)(26,34,31,35)(27,38,39,40)(28,42,41,43)(32,36,45,49)(33,48,46,50)(37,53,52,54)(47,56,55,51)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fgi,d^2=fi,e^2=hi,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^c=ei,e^d=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, 7094566);