#344041

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

606 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)(3,5)(4,6)(7,8)(9,10,12,13)(11,15,16,14)(17,18,22,24,21,26,19,23)(20,28,27,25,30,31,29,32);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,14,13,15)(17,19,21,22)(18,23,26,24)(20,27,30,29)(25,31,32,28);(1,4,5,8)(2,6,3,7)(10,13)(11,15)(14,16)(18,24)(19,22)(23,26)(25,28)(27,29)(31,32);(2,3)(6,7)(11,16)(14,15)(17,20,21,30)(18,25,26,32)(19,27,22,29)(23,31,24,28)

Polycyclic presentation:

2^9|a^2=fh,b^2=eh,c^2=ei,d^2=h,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, 251237);