#226019

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

548 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=h,e^2=h,f^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,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 133215);