#6792315

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1346 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6699511);