#173356

order	512
nilpotency class	3
isoclinism	isoclinic class of #172777
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

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 80552);