#165486

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

671 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,25)(20,30)(21,27)(22,26)(23,28)(24,29)(31,32)(33,34,36,40,37,39,45,47)(35,41,42,48,43,38,44,46);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,23,20)(18,24,28,32)(21,29,22,31)(25,27,30,26)(33,35,37,43)(34,38,39,41)(36,44,45,42)(40,48,47,46);(2,7)(6,14)(8,15)(9,16)(17,20,23,19)(18,25,28,30)(21,29,22,31)(24,27,32,26);(17,21,23,22)(18,26,28,27)(19,29,20,31)(24,25,32,30)(34,39)(38,41)(40,47)(46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gi,d^2=gi,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, 72682);