#170804

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

688 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)(17,18,22,20)(19,25,28,32)(21,29,31,24)(23,30,26,27)(33,34)(35,42)(36,40)(37,45)(38,41)(39,43)(44,47)(46,48);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,12,14,15)(17,19)(18,23)(20,26)(21,30)(22,28)(24,32)(25,29)(27,31);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20,22,18)(19,26,28,23)(21,24,31,29)(25,30,32,27)(33,35)(34,39)(36,43)(37,46)(38,44)(40,47)(41,48)(42,45);(2,8)(6,13)(7,14)(12,15)(18,20)(23,26)(24,29)(25,32)(33,36,38,37)(34,40,41,45)(35,43,44,46)(39,47,48,42)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=i,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, 78000);