#180554

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	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

666 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,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, 87750);