#433291

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1087 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)(3,4)(6,10)(11,12)(14,19,23,22)(15,21,25,24)(16,26)(20,28);(5,6)(7,11)(8,12)(9,10)(13,14,17,22,18,23,27,19)(15,20,21,26,25,28,24,16);(5,7,9,8)(6,11,10,12)(13,15,17,24,18,25,27,21)(14,20,19,26,23,28,22,16);(2,4)(5,8,9,7)(6,11,10,12)(13,16,27,28,18,26,17,20)(14,21,22,25,23,24,19,15)

Polycyclic presentation:

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

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, 340487);