#465257

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=f,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, 372453);