#7181840

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

1330 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)(12,20)(15,21)(19,24)(22,23)(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)(2,6)(4,7)(5,8)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27)(26,31)(28,37)(29,38)(30,36)(32,45)(33,43)(34,42)(35,44)(39,49)(40,51)(41,50)(46,54)(47,53)(48,55)(52,56);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(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)(11,18)(14,17)(19,24)(22,23)(25,28,30,39)(26,33,35,48)(27,37,36,49)(29,40,41,52)(31,43,44,55)(32,46,34,47)(38,51,50,56)(42,53,45,54);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(27,36)(28,39)(31,44)(33,48)(38,50)(40,52)(42,45)(46,47)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=f,d^2=fi,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, 7089036);