#7193398

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7100594);