#94117

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#95
commutator	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂³	#346
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=f,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ei,d^b=dfi

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, 1313);