#96125

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=gi,e^2=g,f^2=gh,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, 3321);