#180628

order	512
nilpotency class	3
isoclinism	isoclinic class of #180581
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	#2733	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

650 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,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, 87824);