#6741979

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1552 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fgi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6649175);