#6469221

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1525 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)(11,15)(13,16)(17,18,21,26,22,27,36,44)(19,28,30,39,31,43,35,23)(20,24,34,40,33,25,41,42)(29,45,47,38,32,46,48,37);(9,10)(11,16)(12,14)(13,15)(17,19,22,31)(18,23,27,39)(20,32,33,29)(21,35,36,30)(24,38,25,37)(26,43,44,28)(34,47,41,48)(40,45,42,46);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20)(18,24)(19,29)(21,34)(22,33)(23,37)(25,27)(26,40)(28,45)(30,47)(31,32)(35,48)(36,41)(38,39)(42,44)(43,46);(10,14)(13,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,38)(24,27)(26,42)(28,46)(30,47)(31,32)(35,48)(36,41)(37,39)(40,44)(43,45);(1,4,5,3)(2,6,7,8)(20,33)(24,25)(29,32)(34,41)(37,38)(40,42)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,e^2=ghi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376417);