#396454

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1274 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)(7,12,8,10)(13,14,19,22,17,16,18,21)(15,23,27,24,25,20,26,28);(5,7,9,8)(6,10,11,12)(13,15,17,25)(14,20,16,23)(18,26,19,27)(21,24,22,28);(13,16,17,14)(15,24,25,28)(18,21,19,22)(20,27,23,26);(2,4)(5,8,9,7)(6,10,11,12)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303650);