#6798655

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
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	#39357	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=fg,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh

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, 6705851);