#127276

order	512
nilpotency class	4
isoclinism	isoclinic class of #7221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

411 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,20,25,23,29,37,28)(19,30,33,44,35,46,31,47)(21,38,36,26,40,27,22,41)(24,34,43,32,45,48,42,39);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,22,34,37,33,40,32,23,35,36,48,20,31,21,39)(18,24,27,44,25,42,41,30,29,45,38,47,28,43,26,46);(17,20,23,37)(18,25,29,28)(19,31,35,33)(21,36,40,22)(24,42,45,43)(26,27,41,38)(30,47,46,44)(32,34,39,48)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=ghi,e^2=i,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,d^b=di,e^b=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34472);