#7180979

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996405
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61424	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=fi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7088175);