#6644729

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^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=dfhi,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, 6551925);