#7193389

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

1652 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)(21,22,28,34)(23,35,40,46)(24,30,25,31)(26,33,27,32)(29,38,47,39)(36,51,41,45)(37,48,52,44)(42,50,43,49);(5,7,9,14)(6,10,13,11)(8,17,18,15)(12,20,16,19)(21,23)(22,29)(24,41)(25,36)(26,39)(27,38)(28,40)(30,48)(31,44)(32,35)(33,46)(34,47)(37,43)(42,52)(45,50)(49,51);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,13)(12,20)(16,19)(17,18)(21,24)(22,30)(23,36)(25,28)(26,43)(27,42)(29,44)(31,34)(32,50)(33,49)(35,51)(37,38)(39,52)(40,41)(45,46)(47,48);(7,14)(10,11)(15,17)(19,20)(21,25,28,24)(22,31,34,30)(23,37,40,52)(26,43,27,42)(29,45,47,51)(32,50,33,49)(35,48,46,44)(36,39,41,38);(5,8,9,18)(6,12,13,16)(7,15,14,17)(10,19,11,20)(21,26,28,27)(22,32,34,33)(23,38,40,39)(24,42,25,43)(29,46,47,35)(30,49,31,50)(36,52,41,37)(44,51,48,45)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=i,e^2=h,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, 7100585);