#401839

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1324 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)(15,21)(16,23)(18,25)(19,27)(22,24)(26,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,14,17,20)(15,18,22,26)(16,19,23,27)(21,25,24,28);(7,11)(9,12)(13,15)(14,18)(16,24)(17,22)(19,28)(20,26)(21,23)(25,27);(1,3,4,2)(16,23)(19,27)(21,24)(25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=e,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, 309035);