#6817400

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1662 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)(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(21,22,27,32)(23,28,35,44)(24,29,34,43)(25,30,36,45)(26,39,41,31)(33,49,51,42)(37,48,40,46)(38,52,50,47);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23,27,35)(22,28,32,44)(24,36,34,25)(26,40,41,37)(29,45,43,30)(31,48,39,46)(33,50,51,38)(42,52,49,47);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,25,23,27,34,36,35)(22,29,30,28,32,43,45,44)(26,40,38,51,41,37,50,33)(31,48,47,49,39,46,52,42);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,25,27,36)(22,30,32,45)(23,24,35,34)(26,38,41,50)(28,29,44,43)(31,47,39,52)(33,37,51,40)(42,46,49,48);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,26,36,50,27,41,25,38)(22,31,45,52,32,39,30,47)(23,33,24,37,35,51,34,40)(28,42,29,46,44,49,43,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgh,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6724596);