#6658593

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632048
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1312 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)(25,26,29,33)(27,35,37,31)(28,39,40,48)(30,43,42,34)(32,44,47,41)(36,51,52,56)(38,46,54,50)(45,53,49,55);(1,2,4,5)(3,6,7,8)(9,11,14,21)(10,15,18,19)(12,22,13,20)(16,24,17,23)(25,27,29,37)(26,31,33,35)(28,36,40,52)(30,38,42,54)(32,45,47,49)(34,46,43,50)(39,56,48,51)(41,53,44,55);(2,5)(6,8)(25,28,29,40)(26,32,33,47)(27,36,37,52)(30,44,42,41)(31,45,35,49)(34,48,43,39)(38,55,54,53)(46,51,50,56);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(27,37)(32,47)(34,43)(36,52)(38,54)(39,48)(53,55);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(26,33)(27,37)(32,47)(34,43)(36,52)(38,54)(39,48)(53,55)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=i,d^2=fi,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=dfh,e^b=ei,g^c=gi

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, 6565789);