#416033

order	512
nilpotency class	3
isoclinism	isoclinic class of #393999
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

1318 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)(7,11)(8,12)(9,10)(13,14)(15,18)(16,24)(17,19)(20,25)(21,28)(22,27)(23,26);(1,2,4,3)(6,10)(11,12)(13,15,17,22)(14,18,19,27)(16,21,25,23)(20,26,24,28);(1,3,4,2)(5,7,9,8)(6,11,10,12)(15,21)(16,25)(18,26)(20,24)(22,23)(27,28);(5,8,9,7)(6,11,10,12)(14,19)(18,27)(20,24)(26,28)

Polycyclic presentation:

2^9|b^2=eh,c^2=e,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 323229);