#458462

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

953 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)(5,6,9,11)(7,10,8,12)(13,14,18,23)(15,16,24,20)(17,26,28,22)(19,27,25,21);(5,7,9,8)(6,10,11,12)(13,15,18,24)(14,19,23,25)(16,22,20,26)(17,27,28,21);(5,8,9,7)(6,10,11,12)(13,15)(14,20)(16,23)(17,27)(18,24)(19,26)(21,28)(22,25);(2,4)(13,16)(14,21)(15,22)(17,19)(18,20)(23,27)(24,26)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365658);