#219563

order	512
nilpotency class	3
isoclinism	isoclinic class of #219552
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	#3205	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126759);