#405774

order	512
nilpotency class	3
isoclinism	isoclinic class of #405774
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14918
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312970);