#180452

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

609 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:

(1,2,7,4)(3,10,14,16)(5,9,6,15)(8,12,11,13)(17,18)(19,25)(20,22)(21,29)(23,28)(24,31)(26,30)(27,32)(33,34)(35,39)(36,41)(37,46)(38,42)(40,43)(44,48)(45,47);(1,3,7,14)(2,8,4,11)(5,13,6,12)(9,16,15,10)(17,19)(18,23)(20,26)(21,30)(22,27)(24,32)(25,29)(28,31)(33,35)(34,39)(36,45)(37,43)(38,44)(40,46)(41,47)(42,48);(1,4,7,2)(3,11,14,8)(5,9,6,15)(10,13,16,12)(17,20,22,18)(19,26,27,23)(21,24,31,29)(25,30,32,28)(33,35)(34,40)(36,43)(37,45)(38,44)(39,46)(41,48)(42,47);(1,5,7,6)(2,9,4,15)(3,12,14,13)(8,10,11,16)(18,20)(19,27)(24,29)(28,30)(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=gh,c^2=g,d^2=ghi,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, 87648);