#455904

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1067 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:

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

Polycyclic presentation:

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

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, 363100);