#7090210

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1640 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(18,23)(19,27)(21,33)(24,38)(25,36)(26,37)(28,42)(30,41)(31,34)(35,40)(39,44)(47,48);(9,10)(11,15)(12,14)(13,16)(17,18,21,23,22,26,33,37)(19,24,29,35,27,40,43,38)(20,30,31,41,32,36,34,25)(28,44,45,48,42,47,46,39);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,22,27)(18,24,26,40)(20,28,32,42)(21,29,33,43)(23,35,37,38)(25,39,41,48)(30,44,36,47)(31,45,34,46);(11,16)(13,15)(17,20,22,32)(18,25,26,41)(19,28,27,42)(21,34,33,31)(23,36,37,30)(24,39,40,48)(29,46,43,45)(35,47,38,44);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,27)(24,40)(28,42)(29,43)(35,38)(39,48)(44,47)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997406);