#7124757

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7031953);