#95323

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	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

476 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)(19,25)(20,28)(21,23)(22,27)(24,30)(26,32)(29,31)(33,34)(35,38,42,47)(36,43)(37,40)(39,45)(41,48,44,46);(1,3,5,14,6,12,4,13)(2,7,9,16,10,15,8,11)(17,19)(18,22)(20,29)(21,26)(23,31)(24,32)(25,28)(27,30)(33,35)(34,38,40,47)(36,44)(37,42)(39,48,43,46)(41,45);(3,12)(7,15)(11,16)(13,14)(18,23)(19,26)(24,28)(27,29)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=gh,f^2=gh,b^a=bd,c^a=ch,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, 2519);