#7034634

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1710 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,d^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,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, 6941830);