#180693

order	512
nilpotency class	3
isoclinism	isoclinic class of #180581
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

635 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)(6,13)(7,12)(9,15)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31)(34,38)(35,41)(36,43)(40,44)(42,45)(46,48);(1,2)(3,6)(4,12)(5,8)(7,11)(9,16)(10,13)(14,15)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32)(33,34,37,40)(35,39,42,47)(36,44,43,38)(41,46,45,48);(1,3,5,10)(2,6,8,13)(4,9,11,15)(7,14,12,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31)(33,35,37,42)(34,39,40,47)(36,45,43,41)(38,46,44,48);(1,4,5,11)(2,7,8,12)(3,9,10,15)(6,14,13,16)(19,26)(22,30)(27,32)(29,31)(35,42)(39,47)(41,45)(46,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,d^b=dhi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87889);