#180304

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

634 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dgi,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, 87500);