#96122

order	512
nilpotency class	3
isoclinism	isoclinic class of #95218
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=g,f^2=gh,b^a=bd,c^a=ch,d^a=dg,f^a=fghi,c^b=cgh,d^b=dh,e^b=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3318);