#6524945

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 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)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22,25,30,26,29,38,43)(23,27,33,40,32,41,48,52)(24,34,35,42,36,44,37,28)(31,45,47,50,46,51,49,39);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,27)(24,31)(25,33)(26,32)(28,39)(29,41)(30,40)(34,45)(35,47)(36,46)(37,49)(38,48)(42,50)(43,52)(44,51);(6,11)(7,13)(8,15)(16,20)(21,24,26,36)(22,28,29,42)(23,31,32,46)(25,37,38,35)(27,39,41,50)(30,44,43,34)(33,49,48,47)(40,51,52,45);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,16,17,20)(22,29)(23,32)(28,42)(30,43)(31,46)(33,48)(34,44)(47,49);(7,13)(10,17)(14,19)(16,20)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6432141);