#7196784

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1736 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,27,28,26,32,25,31)(23,33,41,46,35,29,36,43)(24,37,42,48,40,30,38,44)(34,45,50,51,39,47,49,52);(7,13)(8,16)(10,17)(11,15)(22,28)(24,38)(25,27)(29,43)(31,32)(33,46)(34,49)(36,41)(39,50)(40,42)(44,48)(51,52);(1,2,3,4)(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,23)(22,29)(24,39)(25,36)(26,35)(27,41)(28,43)(30,47)(31,46)(32,33)(34,40)(37,45)(38,50)(42,49)(44,52)(48,51);(7,13)(10,17)(14,19)(18,20)(21,24,25,38,26,40,27,42)(22,30,31,48,32,37,28,44)(23,34,36,49,35,39,41,50)(29,45,46,52,33,47,43,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(23,35)(29,33)(34,39)(36,41)(43,46)(45,47)(49,50)(51,52)

Polycyclic presentation:

2^9|a^2=fhi,c^2=ghi,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7103980);