#7589489

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	#24244	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgi,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7496685);