#441149

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 348345);