#477168

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 384364);