#6654348

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1179 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6561544);