#7160942

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	#45316	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 7068138);