#6558521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1669 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^c=ei,e^d=ehi

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, 6465717);