#7169712

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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1172 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=f,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7076908);