#222272

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
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×#447	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

922 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 129468);