#7019145

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1718 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,28,21,27)(19,29,36,41,31,25,32,38)(20,26,37,39,35,44,33,43)(30,42,46,48,34,40,45,47);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(18,24)(20,33)(21,23)(25,38)(27,28)(29,41)(30,45)(32,36)(34,46)(35,37)(39,43)(47,48);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,34)(21,32)(22,31)(23,36)(24,38)(26,42)(27,41)(28,29)(30,35)(33,46)(37,45)(39,48)(40,44)(43,47);(11,16)(13,15)(17,20,23,37,22,35,21,33)(18,26,24,39,28,44,27,43)(19,30,36,46,31,34,32,45)(25,40,38,47,29,42,41,48);(10,14)(13,15)(19,31)(25,29)(30,34)(32,36)(38,41)(40,42)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6926341);