#7524614

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1434 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)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,26)(21,27)(22,38)(23,28)(24,29)(30,39)(31,40)(32,41)(33,42)(34,43)(35,46)(36,44)(37,45)(47,48);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,24,32)(18,25,29,41)(20,22,31,23)(21,33,37,34)(26,38,40,28)(27,42,45,43)(30,35,47,36)(39,46,48,44);(1,4,5,3)(2,6,7,8)(10,14)(11,15)(17,20,24,31)(18,26,29,40)(19,22,32,23)(21,34,37,33)(25,38,41,28)(27,43,45,42)(30,35,47,36)(39,46,48,44);(10,14)(13,16)(17,21,24,37)(18,27,29,45)(19,30,32,47)(20,33,31,34)(22,36,23,35)(25,39,41,48)(26,42,40,43)(28,46,38,44);(2,7)(6,8)(17,22,24,23)(18,28,29,38)(19,31,32,20)(21,35,37,36)(25,26,41,40)(27,44,45,46)(30,33,47,34)(39,43,48,42)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7431810);