#147189

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	#1934
prime radical	C₂⁴	C₂×SD₁₆
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,h^a=hi,e^c=ei

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, 54385);