#176144

order	512
nilpotency class	3
isoclinism	isoclinic class of #175533
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

631 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 83340);