#96061

order	512
nilpotency class	3
isoclinism	isoclinic class of #95218
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	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

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3257);