#401644

order	512
nilpotency class	3
isoclinism	isoclinic class of #393893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1310 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^c=fi,h^c=hi,f^d=fi

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, 308840);