#441062

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1794 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,c^2=ei,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348258);