#7624451

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624128
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1268 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531647);