#95733

order	512
nilpotency class	3
isoclinism	isoclinic class of #95118
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

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

480 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)(3,11,13,7)(4,8)(5,10)(6,9)(12,16,14,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,34,36,40,37,39,44,47)(35,41,43,48,42,38,45,46);(1,3,5,14,6,13,4,12)(2,7,10,16,9,11,8,15)(17,19)(18,22,24,25)(20,26)(21,27)(23,29,30,31)(28,32)(33,35)(34,38,39,41)(36,43)(37,42)(40,46,47,48)(44,45);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(34,39)(35,42)(40,47)(43,45)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,e^2=i,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fh,c^b=ci,d^b=dh,e^b=eg

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, 2929);