#7529209

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1467 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=i,e^2=f,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, 7436405);