#165536

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=ghi,d^2=g,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh,d^c=dgi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72732);