#6490175

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=h,d^2=hi,e^2=f,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, 6397371);