#465356

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

980 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)(5,6,10,9)(7,12,8,11)(13,14)(15,19)(16,26)(17,23)(18,21)(20,27)(22,24)(25,28);(5,7,10,8)(6,11,9,12)(13,15,24,26,18,20,17,25)(14,19,22,16,21,27,23,28);(5,8,10,7)(6,11,9,12)(13,16,17,27,18,28,24,19)(14,20,22,26,21,15,23,25);(2,4)(5,9,10,6)(7,12,8,11)(14,21)(15,20)(22,23)(25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372552);