#7168548

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

1184 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,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, 7075744);