#7196320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1361 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)(6,8)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26)(27,31)(28,39)(29,33)(30,35)(32,36)(34,38)(37,46)(40,52)(41,48)(42,45)(43,49)(44,50)(47,51)(53,55)(54,56);(1,2)(3,6)(4,5)(7,8)(11,18)(12,21)(14,17)(15,20)(25,27,30,38)(26,31,35,34)(28,36,42,41)(29,37,44,43)(32,45,48,39)(33,46,50,49)(40,51,53,54)(47,55,56,52);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,28,38,41,30,42,27,36)(26,32,31,45,35,48,34,39)(29,40,43,54,44,53,37,51)(33,47,46,55,50,56,49,52);(2,5)(6,8)(11,18)(14,17)(19,24)(22,23)(25,29)(26,33)(27,37)(28,40)(30,44)(31,46)(32,47)(34,49)(35,50)(36,51)(38,43)(39,52)(41,54)(42,53)(45,55)(48,56);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,27,30,38)(26,31,35,34)(28,41,42,36)(29,43,44,37)(32,39,48,45)(33,49,50,46)(40,51,53,54)(47,55,56,52)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fghi,e^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,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7103516);