#6695313

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

1593 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6602509);