#420725

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

1333 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=e,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,f^c=fi,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, 327921);