#180412

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
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	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

665 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=hi,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, 87608);