#175204

order	512
nilpotency class	3
isoclinism	isoclinic class of #174829
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	#2682	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

705 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)(3,9,10,16)(4,12,14,7)(6,13,15,11)(17,18,22,21)(19,23,27,26)(20,24,29,28)(25,31,32,30)(33,34)(35,39)(36,43)(37,45)(38,41)(40,42)(44,47)(46,48);(1,3,5,10)(2,6,8,15)(4,13,14,11)(7,16,12,9)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35,38,37)(34,39,41,45)(36,42,44,46)(40,47,48,43);(17,20,22,29)(18,24,21,28)(19,25,27,32)(23,30,26,31)(33,36,38,44)(34,40,41,48)(35,42,37,46)(39,47,45,43);(3,10)(6,15)(9,16)(11,13)(17,21,22,18)(19,26,27,23)(20,28,29,24)(25,31,32,30)(34,41)(39,45)(40,48)(43,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,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, 82400);