#7182822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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	#3460	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1573 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)(17,18,21,27,22,28,38,26)(19,29,33,23,32,42,48,41)(20,25,36,43,37,45,35,24)(30,40,47,44,34,39,31,46);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,21,33,22,32,38,48)(18,23,26,29,28,41,27,42)(20,34,35,47,37,30,36,31)(24,40,25,44,43,39,45,46);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,22,37)(18,24,28,43)(19,30,32,34)(21,35,38,36)(23,39,41,40)(25,27,45,26)(29,46,42,44)(31,48,47,33);(3,10)(6,9)(11,16)(14,15)(17,20)(18,25)(19,31)(21,36)(22,37)(23,40)(24,26)(27,43)(28,45)(29,46)(30,33)(32,47)(34,48)(35,38)(39,41)(42,44);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(19,32)(23,41)(29,42)(30,34)(31,47)(33,48)(39,40)(44,46)

Polycyclic presentation:

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

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, 7090018);