#412085

order	512
nilpotency class	3
isoclinism	isoclinic class of #394583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|c^2=e,d^2=hi,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,g^c=gi

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, 319281);