#7018720

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	#45236	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1502 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,21,24)(19,29,32,38,30,25,37,40)(20,26,33,43,36,44,35,39)(31,42,46,47,34,41,45,48);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(20,33)(21,22)(25,40)(27,28)(29,38)(31,45)(34,46)(35,36)(39,43)(41,42);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,34)(21,37)(22,32)(23,30)(24,38)(26,42)(27,40)(28,29)(31,36)(33,45)(35,46)(39,48)(41,44)(43,47);(11,16)(13,15)(17,20,22,33,23,36,21,35)(18,26,27,43,28,44,24,39)(19,31,32,46,30,34,37,45)(25,41,40,48,29,42,38,47);(10,14)(13,15)(17,21,23,22)(18,24,28,27)(19,32,30,37)(20,35,36,33)(25,40,29,38)(26,39,44,43)(31,46,34,45)(41,48,42,47)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6925916);