#95575

order	512
nilpotency class	3
isoclinism	isoclinic class of #94978
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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18)(19,26,28,32)(20,24)(21,30)(22,25)(23,27,31,29)(33,34,37,41,38,42,36,40)(35,43,45,47,44,48,46,39);(1,3)(2,7,9,16)(4,13)(5,11)(6,12)(8,10,14,15)(17,19,21,29,22,28,20,27)(18,23,24,26,25,31,30,32)(33,35)(34,39,42,47)(36,45)(37,46)(38,44)(40,43,41,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(35,44)(39,47)(43,48)(45,46)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gh,e^2=i,f^2=hi,b^a=bd,d^a=dg,f^a=fgh,c^b=cgi,e^b=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2771);