#127548

order	512
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=gi,f^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,c^b=ci,d^b=dh,g^b=gi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34744);