#7582398

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1508 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)(19,29,20,25)(21,26,36,45)(22,37,24,27)(30,43,32,39)(31,40,33,41)(34,48,47,42)(35,46,38,44);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,20)(18,25,28,29)(21,34,36,47)(22,31,24,33)(26,42,45,48)(27,40,37,41)(30,38,32,35)(39,44,43,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,23,19)(18,25,28,29)(21,30,36,32)(22,31,24,33)(26,43,45,39)(27,41,37,40)(34,38,47,35)(42,46,48,44);(10,14)(13,15)(17,21,23,36)(18,26,28,45)(19,30,20,32)(22,35,24,38)(25,39,29,43)(27,44,37,46)(31,47,33,34)(40,42,41,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,22,23,24)(18,27,28,37)(19,31,20,33)(21,35,36,38)(25,40,29,41)(26,44,45,46)(30,47,32,34)(39,42,43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7489594);