#170528

order	512
nilpotency class	3
isoclinism	isoclinic class of #170381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

695 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77724);