#6783135

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	#23857	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1632 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)(3,7)(4,6)(5,8)(17,18,23,27)(19,24,29,39)(20,31,35,47)(21,26,22,37)(25,33,43,34)(28,44,32,46)(30,40,36,41)(38,48,42,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,39)(20,32,35,28)(21,36,22,30)(25,42,43,38)(26,41,37,40)(31,46,47,44)(33,45,34,48);(10,13)(14,16)(17,20,22,34,23,35,21,33)(18,25,26,31,27,43,37,47)(19,28,30,45,29,32,36,48)(24,38,41,44,39,42,40,46);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(24,39)(28,32)(30,36)(38,42)(40,41)(44,46)(45,48);(3,8)(5,7)(10,13)(14,16)(17,21,23,22)(18,26,27,37)(19,30,29,36)(20,33,35,34)(24,40,39,41)(25,31,43,47)(28,45,32,48)(38,46,42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=hi,e^2=i,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, 6690331);