#7576606

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1572 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)(3,7)(4,6)(5,8)(10,14)(11,12)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,39)(29,38)(31,37)(32,36)(33,40)(34,41)(35,42)(43,47)(44,46)(45,48);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19)(18,23)(20,32)(21,30)(22,29)(24,39)(25,27)(26,37)(28,33)(31,35)(34,44)(36,40)(38,42)(41,47)(43,45)(46,48);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20,21,33)(18,24,25,40)(19,28,30,32)(22,34,35,45)(23,36,27,39)(26,41,42,48)(29,43,31,44)(37,46,38,47);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,16,14,15)(19,29)(22,35)(23,37)(26,42)(27,38)(28,43)(30,31)(32,44)(34,45)(36,46)(39,47)(41,48);(2,8)(6,7)(9,12,13,11)(10,15,14,16)(18,25)(20,33)(23,27)(26,42)(28,32)(34,45)(37,38)(43,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7483802);