#355211

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂×C₄³

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1410 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)(9,10)(11,15)(12,14)(13,16)(18,23)(19,26)(20,22)(24,31)(27,32)(28,29);(5,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,27)(20,30)(21,23)(22,25)(24,29)(26,31)(28,32);(11,16)(13,15)(17,19,20,28,21,29,22,26)(18,24,30,31,23,27,25,32);(10,14)(13,15)(17,20,21,22)(18,25,23,30)(19,28,29,26)(24,32,27,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fi,d^2=i,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, 262407);