#7532053

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	#45385	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1757 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(24,37)(25,38)(26,35)(30,42)(39,41)(40,46)(43,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,23)(19,24,28,37)(20,30,32,42)(22,35,34,26)(25,36,38,33)(27,43,31,48)(29,46,45,40)(39,47,41,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19,21,28)(18,24,23,37)(20,31,32,27)(22,29,34,45)(25,41,38,39)(26,40,35,46)(30,48,42,43)(33,47,36,44);(10,14)(13,15)(17,20,21,32)(18,25,23,38)(19,27,28,31)(22,36,34,33)(24,39,37,41)(26,42,35,30)(29,47,45,44)(40,48,46,43);(2,5)(6,8)(10,14)(13,15)(19,28)(24,37)(27,31)(29,45)(39,41)(40,46)(43,48)(44,47)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^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,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, 7439249);