#461308

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

965 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368504);