#354503

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,3)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32);(5,6,7,8)(10,14)(15,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(9,11,13,12)(10,15,14,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,3,4,2)(9,12,13,11)(10,16,14,15)(19,26)(22,30)(27,32)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=eg,f^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261699);