#413018

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

740 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,d^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=di,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, 320214);