#411318

order	512
nilpotency class	3
isoclinism	isoclinic class of #393877
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=gi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318514);