#6489102

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1706 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)(9,10)(11,14)(12,15)(13,16)(17,18,24,30)(19,31,35,44)(20,26,21,27)(22,29,23,28)(25,38,41,34)(32,45,33,46)(36,43,37,42)(39,47,40,48);(11,12)(14,15)(17,19)(18,25)(20,33)(21,32)(22,34)(23,38)(24,35)(26,40)(27,39)(28,31)(29,44)(30,41)(36,48)(37,47)(42,45)(43,46);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,20)(18,26)(19,32)(21,24)(22,37)(23,36)(25,39)(27,30)(28,43)(29,42)(31,45)(33,35)(34,48)(38,47)(40,41)(44,46);(9,12,13,11)(10,15,16,14)(17,21,24,20)(18,27,30,26)(19,33,35,32)(22,36,23,37)(25,40,41,39)(28,42,29,43)(31,46,44,45)(34,47,38,48);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(17,22,24,23)(18,28,30,29)(19,34,35,38)(20,36,21,37)(25,31,41,44)(26,42,27,43)(32,47,33,48)(39,46,40,45)

Polycyclic presentation:

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

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, 6396298);