#7199730

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1194 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,21)(14,17)(18,20)(19,23)(22,24)(25,26,29,34,30,35,42,31)(27,32,38,47,37,46,52,43)(28,33,39,49,41,50,54,44)(36,45,51,56,40,48,53,55);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(26,31)(28,39)(29,42)(32,43)(34,35)(36,51)(38,52)(40,53)(41,54)(44,49)(46,47)(55,56);(1,4,5,3)(2,6,7,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,27)(26,32)(28,40)(29,38)(30,37)(31,43)(33,48)(34,47)(35,46)(36,41)(39,53)(42,52)(44,56)(45,50)(49,55)(51,54);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28,29,39,30,41,42,54)(26,33,34,49,35,50,31,44)(27,36,38,51,37,40,52,53)(32,45,47,56,46,48,43,55);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(27,37)(32,46)(36,40)(38,52)(43,47)(45,48)(51,53)(55,56)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=fh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106926);