#170816

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

676 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,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, 78012);