#6412160

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1343 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)(3,5)(4,6)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26,29,33,30,34,41,48)(27,31,37,44,36,43,50,54)(28,32,39,46,40,47,52,56)(35,42,49,53,38,45,51,55);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,18)(22,23)(25,27)(26,31)(28,35)(29,37)(30,36)(32,42)(33,44)(34,43)(38,40)(39,49)(41,50)(45,47)(46,53)(48,54)(51,52)(55,56);(2,6)(3,7)(25,28,29,39,30,40,41,52)(26,32,33,46,34,47,48,56)(27,35,37,49,36,38,50,51)(31,42,44,53,43,45,54,55);(2,6)(5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27)(26,31)(28,38)(29,37)(30,36)(32,45)(33,44)(34,43)(35,40)(39,51)(41,50)(42,47)(46,55)(48,54)(49,52)(53,56);(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(27,36)(31,43)(35,38)(37,50)(42,45)(44,54)(49,51)(53,55)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6319356);