#7044845

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1196 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=fg,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,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, 6952041);