#455333

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1195 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)(6,7)(10,11)(13,14,18,27,16,20,17,21)(15,22,26,23,24,19,25,28);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,24)(17,25)(18,26)(20,22)(21,23)(27,28);(5,7,9,6)(8,10,12,11)(13,14,16,20)(15,23,24,28)(17,27,18,21)(19,26,22,25);(2,4)(5,8,9,12)(6,10,7,11)(14,20)(19,22)(21,27)(23,28)

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=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, 362529);