#96009

order	512
nilpotency class	3
isoclinism	isoclinic class of #95196
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

494 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3205);