#177722

order	512
nilpotency class	3
isoclinism	isoclinic class of #177385
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

674 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=ghi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dghi,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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 84918);