#355534

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

1646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262730);