#147101

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1942
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6,10,17,13,19,12,37)(7,20,24,41,27,39,26,58)(8,28,30,61,33,62,32,45)(9,34,35,47,36,18,11,16)(14,25,40,23,42,57,49,22)(15,43,44,63,46,31,60,29)(21,50,53,65,56,67,55,38)(48,66,64,51,59,54,68,52);(5,7,11,25,12,26,36,22,13,27,35,57,10,24,9,23)(6,14,18,41,17,40,16,39,19,42,34,58,37,49,47,20)(8,21,31,54,32,55,63,51,33,56,43,66,30,53,29,52)(15,38,45,64,44,50,28,59,46,65,61,68,60,67,62,48);(1,2,3,4)(5,8,11,31,12,32,36,63,13,33,35,43,10,30,9,29)(6,15,34,61,17,44,47,62,19,46,18,45,37,60,16,28)(7,21,25,54,26,55,22,51,27,56,57,66,24,53,23,52)(14,38,58,68,40,50,20,48,42,65,41,64,49,67,39,59)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,d^2=hi,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,d^b=di

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, 54297);