#6658525

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632068
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1360 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,g^c=gi,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, 6565721);