#222244

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×#3119	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

929 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:

(1,2,4,7,5,8,11,14)(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)(7,13)(8,12)(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)(17,20,22,34)(18,24,26,32)(19,28,31,45)(21,33,36,48)(23,37,39,46)(25,40,35,47)(27,41,38,42)(29,43,30,44);(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,c^2=h,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, 129440);