#342010

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	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

672 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)(9,10,12,13)(11,15,16,14)(17,18,22,27,23,28,19,24)(20,31,29,26,21,32,30,25);(1,3)(2,5)(4,8)(6,7)(10,13)(11,14)(15,16)(17,19,23,22)(18,24,28,27)(20,29,21,30)(25,31,26,32);(9,11,12,16)(10,14,13,15)(17,20,23,21)(18,25,28,26)(19,29,22,30)(24,31,27,32);(2,6)(3,8)(9,10,12,13)(11,14,16,15)(17,21,23,20)(18,26,28,25)(19,30,22,29)(24,32,27,31)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,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, 249206);