#7184871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1420 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)(3,5,7,8)(10,14)(12,15)(17,18,22,26)(19,24,29,40)(20,31,33,25)(21,23,34,27)(28,42,44,38)(30,36,45,39)(32,37,35,41)(43,47,46,48);(1,3,4,7)(2,5,6,8)(9,10,13,14)(11,12,16,15)(18,23)(20,32)(21,34)(24,36)(25,31)(26,27)(28,43)(30,45)(33,35)(38,42)(39,40)(44,46);(2,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19,22,29)(18,24,26,40)(20,28,33,44)(21,30,34,45)(23,36,27,39)(25,38,31,42)(32,43,35,46)(37,47,41,48);(10,14)(11,16)(17,20,22,33)(18,25,26,31)(19,28,29,44)(21,35,34,32)(23,37,27,41)(24,38,40,42)(30,46,45,43)(36,47,39,48);(9,12,13,15)(10,11,14,16)(18,26)(19,29)(23,27)(25,31)(28,44)(30,45)(37,41)(43,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092067);