#6524523

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1439 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)(3,6)(4,8)(5,7)(17,18,21,26,22,24,34,38)(19,27,29,35,30,40,33,23)(20,25,32,39,31,37,44,47)(28,41,43,46,42,48,45,36);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,24,35)(20,28,31,42)(21,33,34,29)(25,36,37,46)(26,40,38,27)(32,45,44,43)(39,48,47,41);(2,7)(6,8)(10,13)(11,15)(18,24)(23,35)(25,37)(26,38)(27,40)(36,46)(39,47)(41,48);(9,11,12,15)(10,14,13,16)(17,20,22,31)(18,25,24,37)(19,28,30,42)(21,32,34,44)(23,36,35,46)(26,39,38,47)(27,41,40,48)(29,43,33,45);(1,4,5,3)(2,8,7,6)(10,13)(14,16)(20,31)(25,37)(28,42)(32,44)(36,46)(39,47)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6431719);