#6412163

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

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=dgi,e^a=ef,c^b=cgi,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, 6319359);