#321887

order	512
nilpotency class	3
isoclinism	isoclinic class of #279051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

758 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eghi,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,f^c=fi

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, 229083);