#357622

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3114 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:

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

Polycyclic presentation:

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

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, 264818);