#6767969

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#38095	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1756 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)(19,24,29,38)(20,31,34,42)(21,26,22,36)(25,32,41,33)(28,43,44,48)(30,39,35,40)(37,46,47,45);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,38)(20,28,34,44)(21,35,22,30)(25,37,41,47)(26,40,36,39)(31,43,42,48)(32,46,33,45);(10,13)(14,16)(17,20,23,34)(18,25,27,41)(19,28,29,44)(21,33,22,32)(24,37,38,47)(26,42,36,31)(30,46,35,45)(39,43,40,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(24,38)(28,44)(30,35)(37,47)(39,40)(43,48)(45,46);(3,8)(5,7)(10,13)(14,16)(17,21,23,22)(18,26,27,36)(19,30,29,35)(20,32,34,33)(24,39,38,40)(25,31,41,42)(28,45,44,46)(37,48,47,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=h,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=efhi,e^c=ei,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6675165);