#203718

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

595 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)(3,6,9,13,10,12,15,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,44)(31,39)(33,42)(35,41)(37,40)(38,47)(43,46)(45,48);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,38)(26,29)(27,42)(28,39)(34,43)(35,45)(36,46)(37,47)(40,48)(41,44);(17,20,22,36,23,37,21,35)(18,25,26,34,28,41,27,40)(19,30,32,46,33,47,31,45)(24,38,29,43,39,44,42,48);(2,7)(6,12)(8,14)(13,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,39,42)(25,34,41,40)(30,45,47,46)(38,43,44,48)

Polycyclic presentation:

2^9|a^2=g,b^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,e^b=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110914);