#127708

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

536 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)(7,9)(8,11)(10,12)(13,14)(15,26)(16,22)(17,33)(18,23)(19,36)(20,25)(21,28)(24,35)(27,38)(29,37)(30,44)(31,40)(32,41)(34,39)(42,43);(6,9)(7,12)(10,11)(13,15,18,29,19,30,35,28,20,31,32,43,16,27,17,34)(14,21,24,38,22,37,23,40,25,39,33,44,36,42,41,26);(13,16,20,19)(14,22,25,36)(15,27,31,30)(17,32,35,18)(21,37,39,42)(23,33,41,24)(26,38,40,44)(28,29,34,43)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34904);