#222243

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

903 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=de,e^b=eh,g^c=gh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 129439);