#7180971

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	#3035	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 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)(17,18,22,28,24,29,23,41)(19,30,32,45,33,44,40,25)(20,34,35,26,36,46,42,47)(21,37,38,43,39,27,31,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,24,33)(18,25,29,45)(20,31,36,38)(21,35,39,42)(22,40,23,32)(26,43,47,48)(27,46,37,34)(28,44,41,30);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,26)(19,31)(21,32)(22,35)(23,42)(24,36)(25,43)(27,30)(28,46)(29,47)(33,38)(34,41)(37,44)(39,40)(45,48);(3,10)(6,9)(11,16)(14,15)(17,21,22,38,24,39,23,31)(18,27,28,48,29,37,41,43)(19,20,32,35,33,36,40,42)(25,26,30,46,45,47,44,34);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,d^2=f,e^2=h,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, 7088167);