#359128

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1686 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)(6,7)(9,10,12,11)(13,14)(15,18)(16,19)(17,20)(21,25)(22,27)(23,28)(24,26);(5,6)(7,8)(13,15)(14,18)(16,22)(17,24)(19,27)(20,26)(21,28)(23,25);(2,4)(5,7,8,6)(14,19)(15,22)(16,25)(20,21)(23,24)(27,28);(9,11,12,10)(14,20)(18,26)(19,21)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266324);