#95185

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

492 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)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18)(19,22,25,29)(20,27)(21,23)(24,28)(26,32,31,30)(33,34,36,40,37,39,45,47)(35,41,43,48,42,38,44,46);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,20,26,21,25,28,31)(18,22,24,30,23,29,27,32)(33,35)(34,38)(36,44)(37,42)(39,41)(40,48)(43,45)(46,47);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,23)(22,29)(24,27)(30,32)(34,39)(35,42)(40,47)(43,44)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,e^2=i,f^2=h,b^a=bd,c^a=cg,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, 2381);