#342142

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

675 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:

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

Polycyclic presentation:

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

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, 249338);