#180565

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	#2682	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

661 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)(3,7)(4,12)(5,9)(6,11)(8,14)(10,16)(13,15)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,28,32,29)(33,34,38,41)(35,43,44,47)(36,39,37,42)(40,45,48,46);(1,3)(2,6)(4,13)(5,10)(7,12)(8,16)(9,15)(11,14)(17,19,22,30)(18,23,26,32)(20,29,21,28)(24,27,25,31)(34,39)(36,37)(40,47)(41,42)(43,48)(45,46);(1,3)(2,7)(4,11)(5,10)(6,12)(8,15)(9,16)(13,14)(20,21)(24,25)(27,31)(28,29)(33,35)(34,40)(36,46)(37,45)(38,44)(39,47)(41,48)(42,43);(3,10)(6,15)(7,16)(11,13)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27)(34,41)(35,44)(39,42)(45,46)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,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, 87761);