#7182815

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

1617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=i,d^2=i,e^2=h,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, 7090011);