#219690

order	512
nilpotency class	3
isoclinism	isoclinic class of #219638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

475 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dhi,f^b=fh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126886);