#359327

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

946 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266523);